Lazy Series

Coefficients of lazy series are computed on demand. They have infinite precision, although equality can only be decided in special cases.

AUTHORS:

  • Kwankyu Lee (2019-02-24): initial version

  • Tejasvi Chebrolu, Martin Rubey, Travis Scrimshaw (2021-08): refactored and expanded functionality

EXAMPLES:

Laurent series over the integer ring are particularly useful as generating functions for sequences arising in combinatorics.

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)

The generating function of the Fibonacci sequence is:

sage: f = 1 / (1 - z - z^2)
sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
>>> from sage.all import *
>>> f = Integer(1) / (Integer(1) - z - z**Integer(2))
>>> f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)

In principle, we can now compute any coefficient of \(f\):

sage: f.coefficient(100)
573147844013817084101
>>> from sage.all import *
>>> f.coefficient(Integer(100))
573147844013817084101

Which coefficients are actually computed depends on the type of implementation. For the sparse implementation, only the coefficients which are needed are computed.

sage: s = L(lambda n: n, valuation=0); s
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: s.coefficient(10)
10
sage: s._coeff_stream._cache
{1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 10: 10}
>>> from sage.all import *
>>> s = L(lambda n: n, valuation=Integer(0)); s
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
>>> s.coefficient(Integer(10))
10
>>> s._coeff_stream._cache
{1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 10: 10}

Using the dense implementation, all coefficients up to the required coefficient are computed.

sage: L.<x> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: s = L(lambda n: n, valuation=0); s
x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + O(x^7)
sage: s.coefficient(10)
10
sage: s._coeff_stream._cache
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, sparse=False, names=('x',)); (x,) = L._first_ngens(1)
>>> s = L(lambda n: n, valuation=Integer(0)); s
x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + O(x^7)
>>> s.coefficient(Integer(10))
10
>>> s._coeff_stream._cache
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

We can do arithmetic with lazy power series:

sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: f^-1
1 - z - z^2 + O(z^7)
sage: f + f^-1
2 + z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: g = (f + f^-1)*(f - f^-1); g
4*z + 6*z^2 + 8*z^3 + 19*z^4 + 38*z^5 + 71*z^6 + O(z^7)
>>> from sage.all import *
>>> f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
>>> f**-Integer(1)
1 - z - z^2 + O(z^7)
>>> f + f**-Integer(1)
2 + z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
>>> g = (f + f**-Integer(1))*(f - f**-Integer(1)); g
4*z + 6*z^2 + 8*z^3 + 19*z^4 + 38*z^5 + 71*z^6 + O(z^7)

We call lazy power series whose coefficients are known to be eventually constant ‘exact’. In some cases, computations with such series are much faster. Moreover, these are the series where equality can be decided. For example:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: f = 1 + 2*z^2 / (1 - z)
sage: f - 2 / (1 - z) + 1 + 2*z
0
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = Integer(1) + Integer(2)*z**Integer(2) / (Integer(1) - z)
>>> f - Integer(2) / (Integer(1) - z) + Integer(1) + Integer(2)*z
0

However, multivariate Taylor series are actually represented as streams of multivariate polynomials. Therefore, the only exact series in this case are polynomials:

sage: L.<x,y> = LazyPowerSeriesRing(ZZ)
sage: 1 / (1-x)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + O(x,y)^7
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> Integer(1) / (Integer(1)-x)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + O(x,y)^7

A similar statement is true for lazy symmetric functions:

sage: h = SymmetricFunctions(QQ).h()                                                # needs sage.combinat
sage: L = LazySymmetricFunctions(h)                                                 # needs sage.combinat
sage: 1 / (1-L(h[1]))                                                               # needs sage.combinat
h[] + h[1] + (h[1,1]) + (h[1,1,1]) + (h[1,1,1,1]) + (h[1,1,1,1,1]) + (h[1,1,1,1,1,1]) + O^7
>>> from sage.all import *
>>> h = SymmetricFunctions(QQ).h()                                                # needs sage.combinat
>>> L = LazySymmetricFunctions(h)                                                 # needs sage.combinat
>>> Integer(1) / (Integer(1)-L(h[Integer(1)]))                                                               # needs sage.combinat
h[] + h[1] + (h[1,1]) + (h[1,1,1]) + (h[1,1,1,1]) + (h[1,1,1,1,1]) + (h[1,1,1,1,1,1]) + O^7

We can change the base ring:

sage: h = g.change_ring(QQ)
sage: h.parent()                                                                    # needs sage.combinat
Lazy Laurent Series Ring in z over Rational Field
sage: h                                                                             # needs sage.combinat
4*z + 6*z^2 + 8*z^3 + 19*z^4 + 38*z^5 + 71*z^6 + 130*z^7 + O(z^8)
sage: hinv = h^-1; hinv                                                             # needs sage.combinat
1/4*z^-1 - 3/8 + 1/16*z - 17/32*z^2 + 5/64*z^3 - 29/128*z^4 + 165/256*z^5 + O(z^6)
sage: hinv.valuation()                                                              # needs sage.combinat
-1
>>> from sage.all import *
>>> h = g.change_ring(QQ)
>>> h.parent()                                                                    # needs sage.combinat
Lazy Laurent Series Ring in z over Rational Field
>>> h                                                                             # needs sage.combinat
4*z + 6*z^2 + 8*z^3 + 19*z^4 + 38*z^5 + 71*z^6 + 130*z^7 + O(z^8)
>>> hinv = h**-Integer(1); hinv                                                             # needs sage.combinat
1/4*z^-1 - 3/8 + 1/16*z - 17/32*z^2 + 5/64*z^3 - 29/128*z^4 + 165/256*z^5 + O(z^6)
>>> hinv.valuation()                                                              # needs sage.combinat
-1
class sage.rings.lazy_series.LazyCauchyProductSeries(parent, coeff_stream)[source]

Bases: LazyModuleElement

A class for series where multiplication is the Cauchy product.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1 / (1 - z)
sage: f
1 + z + z^2 + O(z^3)
sage: f * (1 - z)
1

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: f = 1 / (1 - z)
sage: f
1 + z + z^2 + O(z^3)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = Integer(1) / (Integer(1) - z)
>>> f
1 + z + z^2 + O(z^3)
>>> f * (Integer(1) - z)
1

>>> L = LazyLaurentSeriesRing(ZZ, sparse=True, names=('z',)); (z,) = L._first_ngens(1)
>>> f = Integer(1) / (Integer(1) - z)
>>> f
1 + z + z^2 + O(z^3)
exp()[source]

Return the exponential series of self.

We use the identity

\[\exp(s) = 1 + \int s' \exp(s).\]

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: exp(z)
1 + z + 1/2*z^2 + 1/6*z^3 + 1/24*z^4 + 1/120*z^5 + 1/720*z^6 + O(z^7)
sage: exp(z + z^2)
1 + z + 3/2*z^2 + 7/6*z^3 + 25/24*z^4 + 27/40*z^5 + 331/720*z^6 + O(z^7)
sage: exp(0)                                                                # needs sage.symbolic
1
sage: exp(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: exp(x+y)[4].factor()
(1/24) * (x + y)^4
sage: exp(x/(1-y)).polynomial(3)
1/6*x^3 + x^2*y + x*y^2 + 1/2*x^2 + x*y + x + 1
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> exp(z)
1 + z + 1/2*z^2 + 1/6*z^3 + 1/24*z^4 + 1/120*z^5 + 1/720*z^6 + O(z^7)
>>> exp(z + z**Integer(2))
1 + z + 3/2*z^2 + 7/6*z^3 + 25/24*z^4 + 27/40*z^5 + 331/720*z^6 + O(z^7)
>>> exp(Integer(0))                                                                # needs sage.symbolic
1
>>> exp(Integer(1) + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> exp(x+y)[Integer(4)].factor()
(1/24) * (x + y)^4
>>> exp(x/(Integer(1)-y)).polynomial(Integer(3))
1/6*x^3 + x^2*y + x*y^2 + 1/2*x^2 + x*y + x + 1
log()[source]

Return the series for the natural logarithm of self.

We use the identity

\[\log(s) = \int s' / s.\]

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: log(1/(1-z))
z + 1/2*z^2 + 1/3*z^3 + 1/4*z^4 + 1/5*z^5 + 1/6*z^6 + 1/7*z^7 + O(z^8)

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: log((1 + x/(1-y))).polynomial(3)
1/3*x^3 - x^2*y + x*y^2 - 1/2*x^2 + x*y + x
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> log(Integer(1)/(Integer(1)-z))
z + 1/2*z^2 + 1/3*z^3 + 1/4*z^4 + 1/5*z^5 + 1/6*z^6 + 1/7*z^7 + O(z^8)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> log((Integer(1) + x/(Integer(1)-y))).polynomial(Integer(3))
1/3*x^3 - x^2*y + x*y^2 - 1/2*x^2 + x*y + x
valuation()[source]

Return the valuation of self.

This method determines the valuation of the series by looking for a nonzero coefficient. Hence if the series happens to be zero, then it may run forever.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: s = 1/(1 - z) - 1/(1 - 2*z)
sage: s.valuation()
1
sage: t = z - z
sage: t.valuation()
+Infinity
sage: M = L(lambda n: n^2, 0)
sage: M.valuation()
1
sage: (M - M).valuation()
+Infinity
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> s = Integer(1)/(Integer(1) - z) - Integer(1)/(Integer(1) - Integer(2)*z)
>>> s.valuation()
1
>>> t = z - z
>>> t.valuation()
+Infinity
>>> M = L(lambda n: n**Integer(2), Integer(0))
>>> M.valuation()
1
>>> (M - M).valuation()
+Infinity
class sage.rings.lazy_series.LazyCompletionGradedAlgebraElement(parent, coeff_stream)[source]

Bases: LazyCauchyProductSeries

An element of a completion of a graded algebra that is computed lazily.

class sage.rings.lazy_series.LazyDirichletSeries(parent, coeff_stream)[source]

Bases: LazyModuleElement

A Dirichlet series where the coefficients are computed lazily.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: f = L(constant=1)^2
sage: f                                                                         # needs sage.symbolic
1 + 2/2^z + 2/3^z + 3/4^z + 2/5^z + 4/6^z + 2/7^z + O(1/(8^z))
sage: f.coefficient(100) == number_of_divisors(100)                             # needs sage.libs.pari
True
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, "z")
>>> f = L(constant=Integer(1))**Integer(2)
>>> f                                                                         # needs sage.symbolic
1 + 2/2^z + 2/3^z + 3/4^z + 2/5^z + 4/6^z + 2/7^z + O(1/(8^z))
>>> f.coefficient(Integer(100)) == number_of_divisors(Integer(100))                             # needs sage.libs.pari
True

Lazy Dirichlet series is picklable:

sage: g = loads(dumps(f))
sage: g                                                                         # needs sage.symbolic
1 + 2/2^z + 2/3^z + 3/4^z + 2/5^z + 4/6^z + 2/7^z + O(1/(8^z))
sage: g == f
True
>>> from sage.all import *
>>> g = loads(dumps(f))
>>> g                                                                         # needs sage.symbolic
1 + 2/2^z + 2/3^z + 3/4^z + 2/5^z + 4/6^z + 2/7^z + O(1/(8^z))
>>> g == f
True
is_unit()[source]

Return whether this element is a unit in the ring.

EXAMPLES:

sage: D = LazyDirichletSeriesRing(ZZ, "s")
sage: D([0, 2]).is_unit()
False

sage: D([-1, 2]).is_unit()
True

sage: D([3, 2]).is_unit()
False

sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: D([3, 2]).is_unit()
True
>>> from sage.all import *
>>> D = LazyDirichletSeriesRing(ZZ, "s")
>>> D([Integer(0), Integer(2)]).is_unit()
False

>>> D([-Integer(1), Integer(2)]).is_unit()
True

>>> D([Integer(3), Integer(2)]).is_unit()
False

>>> D = LazyDirichletSeriesRing(QQ, "s")
>>> D([Integer(3), Integer(2)]).is_unit()
True
valuation()[source]

Return the valuation of self.

This method determines the valuation of the series by looking for a nonzero coefficient. Hence if the series happens to be zero, then it may run forever.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: mu = L(moebius); mu.valuation()                                       # needs sage.libs.pari
0
sage: (mu - mu).valuation()                                                 # needs sage.libs.pari
+Infinity
sage: g = L(constant=1, valuation=2)
sage: g.valuation()                                                         # needs sage.symbolic
log(2)
sage: (g*g).valuation()                                                     # needs sage.symbolic
2*log(2)
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, "z")
>>> mu = L(moebius); mu.valuation()                                       # needs sage.libs.pari
0
>>> (mu - mu).valuation()                                                 # needs sage.libs.pari
+Infinity
>>> g = L(constant=Integer(1), valuation=Integer(2))
>>> g.valuation()                                                         # needs sage.symbolic
log(2)
>>> (g*g).valuation()                                                     # needs sage.symbolic
2*log(2)
class sage.rings.lazy_series.LazyLaurentSeries(parent, coeff_stream)[source]

Bases: LazyCauchyProductSeries

A Laurent series where the coefficients are computed lazily.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)

We can build a series from a function and specify if the series eventually takes a constant value:

sage: f = L(lambda i: i, valuation=-3, constant=-1, degree=3)
sage: f
-3*z^-3 - 2*z^-2 - z^-1 + z + 2*z^2 - z^3 - z^4 - z^5 + O(z^6)
sage: f[-2]
-2
sage: f[10]
-1
sage: f[-5]
0

sage: f = L(lambda i: i, valuation=-3)
sage: f
-3*z^-3 - 2*z^-2 - z^-1 + z + 2*z^2 + 3*z^3 + O(z^4)
sage: f[20]
20
>>> from sage.all import *
>>> f = L(lambda i: i, valuation=-Integer(3), constant=-Integer(1), degree=Integer(3))
>>> f
-3*z^-3 - 2*z^-2 - z^-1 + z + 2*z^2 - z^3 - z^4 - z^5 + O(z^6)
>>> f[-Integer(2)]
-2
>>> f[Integer(10)]
-1
>>> f[-Integer(5)]
0

>>> f = L(lambda i: i, valuation=-Integer(3))
>>> f
-3*z^-3 - 2*z^-2 - z^-1 + z + 2*z^2 + 3*z^3 + O(z^4)
>>> f[Integer(20)]
20

Anything that converts into a polynomial can be input, where we can also specify the valuation or if the series eventually takes a constant value:

sage: L([-5,2,0,5])
-5 + 2*z + 5*z^3
sage: L([-5,2,0,5], constant=6)
-5 + 2*z + 5*z^3 + 6*z^4 + 6*z^5 + 6*z^6 + O(z^7)
sage: L([-5,2,0,5], degree=6, constant=6)
-5 + 2*z + 5*z^3 + 6*z^6 + 6*z^7 + 6*z^8 + O(z^9)
sage: L([-5,2,0,5], valuation=-2, degree=3, constant=6)
-5*z^-2 + 2*z^-1 + 5*z + 6*z^3 + 6*z^4 + 6*z^5 + O(z^6)
sage: L([-5,2,0,5], valuation=5)
-5*z^5 + 2*z^6 + 5*z^8
sage: L({-2:9, 3:4}, constant=2, degree=5)
9*z^-2 + 4*z^3 + 2*z^5 + 2*z^6 + 2*z^7 + O(z^8)
>>> from sage.all import *
>>> L([-Integer(5),Integer(2),Integer(0),Integer(5)])
-5 + 2*z + 5*z^3
>>> L([-Integer(5),Integer(2),Integer(0),Integer(5)], constant=Integer(6))
-5 + 2*z + 5*z^3 + 6*z^4 + 6*z^5 + 6*z^6 + O(z^7)
>>> L([-Integer(5),Integer(2),Integer(0),Integer(5)], degree=Integer(6), constant=Integer(6))
-5 + 2*z + 5*z^3 + 6*z^6 + 6*z^7 + 6*z^8 + O(z^9)
>>> L([-Integer(5),Integer(2),Integer(0),Integer(5)], valuation=-Integer(2), degree=Integer(3), constant=Integer(6))
-5*z^-2 + 2*z^-1 + 5*z + 6*z^3 + 6*z^4 + 6*z^5 + O(z^6)
>>> L([-Integer(5),Integer(2),Integer(0),Integer(5)], valuation=Integer(5))
-5*z^5 + 2*z^6 + 5*z^8
>>> L({-Integer(2):Integer(9), Integer(3):Integer(4)}, constant=Integer(2), degree=Integer(5))
9*z^-2 + 4*z^3 + 2*z^5 + 2*z^6 + 2*z^7 + O(z^8)

We can also perform arithmetic:

sage: f = 1 / (1 - z - z^2)
sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: f.coefficient(100)
573147844013817084101
sage: f = (z^-2 - 1 + 2*z) / (z^-1 - z + 3*z^2)
sage: f
z^-1 - z^2 - z^4 + 3*z^5 + O(z^6)
>>> from sage.all import *
>>> f = Integer(1) / (Integer(1) - z - z**Integer(2))
>>> f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
>>> f.coefficient(Integer(100))
573147844013817084101
>>> f = (z**-Integer(2) - Integer(1) + Integer(2)*z) / (z**-Integer(1) - z + Integer(3)*z**Integer(2))
>>> f
z^-1 - z^2 - z^4 + 3*z^5 + O(z^6)

However, we may not always be able to know when a result is exactly a polynomial:

sage: f * (z^-1 - z + 3*z^2)
z^-2 - 1 + 2*z + O(z^5)
>>> from sage.all import *
>>> f * (z**-Integer(1) - z + Integer(3)*z**Integer(2))
z^-2 - 1 + 2*z + O(z^5)
O(prec, name=None)[source]

Return the Laurent series with absolute precision prec approximated from this series.

INPUT:

  • prec – integer

  • name – name of the variable; if it is None, the name of the variable of the series is used

OUTPUT: a Laurent series with absolute precision prec

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: z = L.gen()
sage: f = (z - 2*z^3)^5/(1 - 2*z)
sage: f
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + 32*z^10 - 16*z^11 + O(z^12)
sage: g = f.approximate_series(10)
sage: g
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + O(z^10)
sage: g.parent()
Power Series Ring in z over Integer Ring
sage: h = (f^-1).approximate_series(3)
sage: h
z^-5 - 2*z^-4 + 10*z^-3 - 20*z^-2 + 60*z^-1 - 120 + 280*z - 560*z^2 + O(z^3)
sage: h.parent()
Laurent Series Ring in z over Integer Ring
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> z = L.gen()
>>> f = (z - Integer(2)*z**Integer(3))**Integer(5)/(Integer(1) - Integer(2)*z)
>>> f
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + 32*z^10 - 16*z^11 + O(z^12)
>>> g = f.approximate_series(Integer(10))
>>> g
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + O(z^10)
>>> g.parent()
Power Series Ring in z over Integer Ring
>>> h = (f**-Integer(1)).approximate_series(Integer(3))
>>> h
z^-5 - 2*z^-4 + 10*z^-3 - 20*z^-2 + 60*z^-1 - 120 + 280*z - 560*z^2 + O(z^3)
>>> h.parent()
Laurent Series Ring in z over Integer Ring
add_bigoh(prec, name=None)[source]

Return the Laurent series with absolute precision prec approximated from this series.

INPUT:

  • prec – integer

  • name – name of the variable; if it is None, the name of the variable of the series is used

OUTPUT: a Laurent series with absolute precision prec

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: z = L.gen()
sage: f = (z - 2*z^3)^5/(1 - 2*z)
sage: f
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + 32*z^10 - 16*z^11 + O(z^12)
sage: g = f.approximate_series(10)
sage: g
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + O(z^10)
sage: g.parent()
Power Series Ring in z over Integer Ring
sage: h = (f^-1).approximate_series(3)
sage: h
z^-5 - 2*z^-4 + 10*z^-3 - 20*z^-2 + 60*z^-1 - 120 + 280*z - 560*z^2 + O(z^3)
sage: h.parent()
Laurent Series Ring in z over Integer Ring
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> z = L.gen()
>>> f = (z - Integer(2)*z**Integer(3))**Integer(5)/(Integer(1) - Integer(2)*z)
>>> f
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + 32*z^10 - 16*z^11 + O(z^12)
>>> g = f.approximate_series(Integer(10))
>>> g
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + O(z^10)
>>> g.parent()
Power Series Ring in z over Integer Ring
>>> h = (f**-Integer(1)).approximate_series(Integer(3))
>>> h
z^-5 - 2*z^-4 + 10*z^-3 - 20*z^-2 + 60*z^-1 - 120 + 280*z - 560*z^2 + O(z^3)
>>> h.parent()
Laurent Series Ring in z over Integer Ring
approximate_series(prec, name=None)[source]

Return the Laurent series with absolute precision prec approximated from this series.

INPUT:

  • prec – integer

  • name – name of the variable; if it is None, the name of the variable of the series is used

OUTPUT: a Laurent series with absolute precision prec

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: z = L.gen()
sage: f = (z - 2*z^3)^5/(1 - 2*z)
sage: f
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + 32*z^10 - 16*z^11 + O(z^12)
sage: g = f.approximate_series(10)
sage: g
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + O(z^10)
sage: g.parent()
Power Series Ring in z over Integer Ring
sage: h = (f^-1).approximate_series(3)
sage: h
z^-5 - 2*z^-4 + 10*z^-3 - 20*z^-2 + 60*z^-1 - 120 + 280*z - 560*z^2 + O(z^3)
sage: h.parent()
Laurent Series Ring in z over Integer Ring
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> z = L.gen()
>>> f = (z - Integer(2)*z**Integer(3))**Integer(5)/(Integer(1) - Integer(2)*z)
>>> f
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + 32*z^10 - 16*z^11 + O(z^12)
>>> g = f.approximate_series(Integer(10))
>>> g
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + O(z^10)
>>> g.parent()
Power Series Ring in z over Integer Ring
>>> h = (f**-Integer(1)).approximate_series(Integer(3))
>>> h
z^-5 - 2*z^-4 + 10*z^-3 - 20*z^-2 + 60*z^-1 - 120 + 280*z - 560*z^2 + O(z^3)
>>> h.parent()
Laurent Series Ring in z over Integer Ring
compose(g)[source]

Return the composition of self with g.

Given two Laurent series \(f\) and \(g\) over the same base ring, the composition \((f \circ g)(z) = f(g(z))\) is defined if and only if:

  • \(g = 0\) and \(\mathrm{val}(f) \geq 0\),

  • \(g\) is nonzero and \(f\) has only finitely many nonzero coefficients,

  • \(g\) is nonzero and \(\mathrm{val}(g) > 0\).

INPUT:

  • g – other series

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = z^2 + 1 + z
sage: f(0)
1
sage: f(L(0))
1
sage: f(f)
3 + 3*z + 4*z^2 + 2*z^3 + z^4
sage: g = z^-3/(1-2*z); g
z^-3 + 2*z^-2 + 4*z^-1 + 8 + 16*z + 32*z^2 + 64*z^3 + O(z^4)
sage: f(g)
z^-6 + 4*z^-5 + 12*z^-4 + 33*z^-3 + 82*z^-2 + 196*z^-1 + 457 + O(z)
sage: g^2 + 1 + g
z^-6 + 4*z^-5 + 12*z^-4 + 33*z^-3 + 82*z^-2 + 196*z^-1 + 457 + O(z)
sage: f(int(2))
7

sage: f = z^-2 + z + 4*z^3
sage: f(f)
4*z^-6 + 12*z^-3 + z^-2 + 48*z^-1 + 12 + O(z)
sage: f^-2 + f + 4*f^3
4*z^-6 + 12*z^-3 + z^-2 + 48*z^-1 + 12 + O(z)
sage: f(g)
4*z^-9 + 24*z^-8 + 96*z^-7 + 320*z^-6 + 960*z^-5 + 2688*z^-4 + 7169*z^-3 + O(z^-2)
sage: g^-2 + g + 4*g^3
4*z^-9 + 24*z^-8 + 96*z^-7 + 320*z^-6 + 960*z^-5 + 2688*z^-4 + 7169*z^-3 + O(z^-2)

sage: f = z^-3 + z^-2 + 1 / (1 + z^2); f
z^-3 + z^-2 + 1 - z^2 + O(z^4)
sage: g = z^3 / (1 + z - z^3); g
z^3 - z^4 + z^5 - z^7 + 2*z^8 - 2*z^9 + O(z^10)
sage: f(g)
z^-9 + 3*z^-8 + 3*z^-7 - z^-6 - 4*z^-5 - 2*z^-4 + z^-3 + O(z^-2)
sage: g^-3 + g^-2 + 1 / (1 + g^2)
z^-9 + 3*z^-8 + 3*z^-7 - z^-6 - 4*z^-5 - 2*z^-4 + z^-3 + O(z^-2)

sage: f = z^-3
sage: g = z^-2 + z^-1
sage: g^(-3)
z^6 - 3*z^7 + 6*z^8 - 10*z^9 + 15*z^10 - 21*z^11 + 28*z^12 + O(z^13)
sage: f(g)
z^6 - 3*z^7 + 6*z^8 - 10*z^9 + 15*z^10 - 21*z^11 + 28*z^12 + O(z^13)

sage: f = z^2 + z^3
sage: g = z^-3 + z^-2
sage: f^-3 + f^-2
z^-6 - 3*z^-5 + 7*z^-4 - 12*z^-3 + 18*z^-2 - 25*z^-1 + 33 + O(z)
sage: g(f)
z^-6 - 3*z^-5 + 7*z^-4 - 12*z^-3 + 18*z^-2 - 25*z^-1 + 33 + O(z)
sage: g^2 + g^3
z^-9 + 3*z^-8 + 3*z^-7 + 2*z^-6 + 2*z^-5 + z^-4
sage: f(g)
z^-9 + 3*z^-8 + 3*z^-7 + 2*z^-6 + 2*z^-5 + z^-4

sage: f = L(lambda n: n, valuation=0); f
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: f(z^2)
z^2 + 2*z^4 + 3*z^6 + 4*z^8 + O(z^9)

sage: f = L(lambda n: n, valuation=-2); f
-2*z^-2 - z^-1 + z + 2*z^2 + 3*z^3 + 4*z^4 + O(z^5)
sage: f3 = f(z^3); f3
-2*z^-6 - z^-3 + O(z)
sage: [f3[i] for i in range(-6,13)]
[-2, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4]
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = z**Integer(2) + Integer(1) + z
>>> f(Integer(0))
1
>>> f(L(Integer(0)))
1
>>> f(f)
3 + 3*z + 4*z^2 + 2*z^3 + z^4
>>> g = z**-Integer(3)/(Integer(1)-Integer(2)*z); g
z^-3 + 2*z^-2 + 4*z^-1 + 8 + 16*z + 32*z^2 + 64*z^3 + O(z^4)
>>> f(g)
z^-6 + 4*z^-5 + 12*z^-4 + 33*z^-3 + 82*z^-2 + 196*z^-1 + 457 + O(z)
>>> g**Integer(2) + Integer(1) + g
z^-6 + 4*z^-5 + 12*z^-4 + 33*z^-3 + 82*z^-2 + 196*z^-1 + 457 + O(z)
>>> f(int(Integer(2)))
7

>>> f = z**-Integer(2) + z + Integer(4)*z**Integer(3)
>>> f(f)
4*z^-6 + 12*z^-3 + z^-2 + 48*z^-1 + 12 + O(z)
>>> f**-Integer(2) + f + Integer(4)*f**Integer(3)
4*z^-6 + 12*z^-3 + z^-2 + 48*z^-1 + 12 + O(z)
>>> f(g)
4*z^-9 + 24*z^-8 + 96*z^-7 + 320*z^-6 + 960*z^-5 + 2688*z^-4 + 7169*z^-3 + O(z^-2)
>>> g**-Integer(2) + g + Integer(4)*g**Integer(3)
4*z^-9 + 24*z^-8 + 96*z^-7 + 320*z^-6 + 960*z^-5 + 2688*z^-4 + 7169*z^-3 + O(z^-2)

>>> f = z**-Integer(3) + z**-Integer(2) + Integer(1) / (Integer(1) + z**Integer(2)); f
z^-3 + z^-2 + 1 - z^2 + O(z^4)
>>> g = z**Integer(3) / (Integer(1) + z - z**Integer(3)); g
z^3 - z^4 + z^5 - z^7 + 2*z^8 - 2*z^9 + O(z^10)
>>> f(g)
z^-9 + 3*z^-8 + 3*z^-7 - z^-6 - 4*z^-5 - 2*z^-4 + z^-3 + O(z^-2)
>>> g**-Integer(3) + g**-Integer(2) + Integer(1) / (Integer(1) + g**Integer(2))
z^-9 + 3*z^-8 + 3*z^-7 - z^-6 - 4*z^-5 - 2*z^-4 + z^-3 + O(z^-2)

>>> f = z**-Integer(3)
>>> g = z**-Integer(2) + z**-Integer(1)
>>> g**(-Integer(3))
z^6 - 3*z^7 + 6*z^8 - 10*z^9 + 15*z^10 - 21*z^11 + 28*z^12 + O(z^13)
>>> f(g)
z^6 - 3*z^7 + 6*z^8 - 10*z^9 + 15*z^10 - 21*z^11 + 28*z^12 + O(z^13)

>>> f = z**Integer(2) + z**Integer(3)
>>> g = z**-Integer(3) + z**-Integer(2)
>>> f**-Integer(3) + f**-Integer(2)
z^-6 - 3*z^-5 + 7*z^-4 - 12*z^-3 + 18*z^-2 - 25*z^-1 + 33 + O(z)
>>> g(f)
z^-6 - 3*z^-5 + 7*z^-4 - 12*z^-3 + 18*z^-2 - 25*z^-1 + 33 + O(z)
>>> g**Integer(2) + g**Integer(3)
z^-9 + 3*z^-8 + 3*z^-7 + 2*z^-6 + 2*z^-5 + z^-4
>>> f(g)
z^-9 + 3*z^-8 + 3*z^-7 + 2*z^-6 + 2*z^-5 + z^-4

>>> f = L(lambda n: n, valuation=Integer(0)); f
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
>>> f(z**Integer(2))
z^2 + 2*z^4 + 3*z^6 + 4*z^8 + O(z^9)

>>> f = L(lambda n: n, valuation=-Integer(2)); f
-2*z^-2 - z^-1 + z + 2*z^2 + 3*z^3 + 4*z^4 + O(z^5)
>>> f3 = f(z**Integer(3)); f3
-2*z^-6 - z^-3 + O(z)
>>> [f3[i] for i in range(-Integer(6),Integer(13))]
[-2, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4]

We compose a Laurent polynomial with a generic element:

sage: R.<x> = QQ[]
sage: f = z^2 + 1 + z^-1
sage: g = x^2 + x + 3
sage: f(g)
(x^6 + 3*x^5 + 12*x^4 + 19*x^3 + 37*x^2 + 28*x + 31)/(x^2 + x + 3)
sage: f(g) == g^2 + 1 + g^-1
True
>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = z**Integer(2) + Integer(1) + z**-Integer(1)
>>> g = x**Integer(2) + x + Integer(3)
>>> f(g)
(x^6 + 3*x^5 + 12*x^4 + 19*x^3 + 37*x^2 + 28*x + 31)/(x^2 + x + 3)
>>> f(g) == g**Integer(2) + Integer(1) + g**-Integer(1)
True

We compose with another lazy Laurent series:

sage: LS.<y> = LazyLaurentSeriesRing(QQ)
sage: f = z^2 + 1 + z^-1
sage: fy = f(y); fy
y^-1 + 1 + y^2
sage: fy.parent() is LS
True
sage: g = y - y
sage: f(g)
Traceback (most recent call last):
...
ZeroDivisionError: the valuation of the series must be nonnegative

sage: g = 1 - y
sage: f(g)
3 - y + 2*y^2 + y^3 + y^4 + y^5 + O(y^6)
sage: g^2 + 1 + g^-1
3 - y + 2*y^2 + y^3 + y^4 + y^5 + O(y^6)

sage: f = L(lambda n: n, valuation=0); f
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: f(0)
0
sage: f(y)
y + 2*y^2 + 3*y^3 + 4*y^4 + 5*y^5 + 6*y^6 + 7*y^7 + O(y^8)
sage: fp = f(y - y)
sage: fp == 0
True
sage: fp.parent() is LS
True

sage: f = z^2 + 3 + z
sage: f(y - y)
3
>>> from sage.all import *
>>> LS = LazyLaurentSeriesRing(QQ, names=('y',)); (y,) = LS._first_ngens(1)
>>> f = z**Integer(2) + Integer(1) + z**-Integer(1)
>>> fy = f(y); fy
y^-1 + 1 + y^2
>>> fy.parent() is LS
True
>>> g = y - y
>>> f(g)
Traceback (most recent call last):
...
ZeroDivisionError: the valuation of the series must be nonnegative

>>> g = Integer(1) - y
>>> f(g)
3 - y + 2*y^2 + y^3 + y^4 + y^5 + O(y^6)
>>> g**Integer(2) + Integer(1) + g**-Integer(1)
3 - y + 2*y^2 + y^3 + y^4 + y^5 + O(y^6)

>>> f = L(lambda n: n, valuation=Integer(0)); f
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
>>> f(Integer(0))
0
>>> f(y)
y + 2*y^2 + 3*y^3 + 4*y^4 + 5*y^5 + 6*y^6 + 7*y^7 + O(y^8)
>>> fp = f(y - y)
>>> fp == Integer(0)
True
>>> fp.parent() is LS
True

>>> f = z**Integer(2) + Integer(3) + z
>>> f(y - y)
3

With both of them sparse:

sage: L.<z> = LazyLaurentSeriesRing(QQ, sparse=True)
sage: LS.<y> = LazyLaurentSeriesRing(QQ, sparse=True)
sage: f = L(lambda n: 1, valuation=0); f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: f(y^2)
1 + y^2 + y^4 + y^6 + O(y^7)

sage: fp = f - 1 + z^-2; fp
z^-2 + z + z^2 + z^3 + z^4 + O(z^5)
sage: fpy = fp(y^2); fpy
y^-4 + y^2 + O(y^3)
sage: fpy.parent() is LS
True
sage: [fpy[i] for i in range(-4,11)]
[1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

sage: g = LS(valuation=2, constant=1); g
y^2 + y^3 + y^4 + O(y^5)
sage: fg = f(g); fg
1 + y^2 + y^3 + 2*y^4 + 3*y^5 + 5*y^6 + O(y^7)
sage: 1 + g + g^2 + g^3 + g^4 + g^5 + g^6
1 + y^2 + y^3 + 2*y^4 + 3*y^5 + 5*y^6 + O(y^7)

sage: h = LS(lambda n: 1 if n % 2 else 0, valuation=2); h
y^3 + y^5 + y^7 + O(y^9)
sage: fgh = fg(h); fgh
1 + y^6 + O(y^7)
sage: [fgh[i] for i in range(0, 15)]
[1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 6, 13]
sage: t = 1 + h^2 + h^3 + 2*h^4 + 3*h^5 + 5*h^6
sage: [t[i] for i in range(0, 15)]
[1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 6, 13]
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, sparse=True, names=('z',)); (z,) = L._first_ngens(1)
>>> LS = LazyLaurentSeriesRing(QQ, sparse=True, names=('y',)); (y,) = LS._first_ngens(1)
>>> f = L(lambda n: Integer(1), valuation=Integer(0)); f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
>>> f(y**Integer(2))
1 + y^2 + y^4 + y^6 + O(y^7)

>>> fp = f - Integer(1) + z**-Integer(2); fp
z^-2 + z + z^2 + z^3 + z^4 + O(z^5)
>>> fpy = fp(y**Integer(2)); fpy
y^-4 + y^2 + O(y^3)
>>> fpy.parent() is LS
True
>>> [fpy[i] for i in range(-Integer(4),Integer(11))]
[1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

>>> g = LS(valuation=Integer(2), constant=Integer(1)); g
y^2 + y^3 + y^4 + O(y^5)
>>> fg = f(g); fg
1 + y^2 + y^3 + 2*y^4 + 3*y^5 + 5*y^6 + O(y^7)
>>> Integer(1) + g + g**Integer(2) + g**Integer(3) + g**Integer(4) + g**Integer(5) + g**Integer(6)
1 + y^2 + y^3 + 2*y^4 + 3*y^5 + 5*y^6 + O(y^7)

>>> h = LS(lambda n: Integer(1) if n % Integer(2) else Integer(0), valuation=Integer(2)); h
y^3 + y^5 + y^7 + O(y^9)
>>> fgh = fg(h); fgh
1 + y^6 + O(y^7)
>>> [fgh[i] for i in range(Integer(0), Integer(15))]
[1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 6, 13]
>>> t = Integer(1) + h**Integer(2) + h**Integer(3) + Integer(2)*h**Integer(4) + Integer(3)*h**Integer(5) + Integer(5)*h**Integer(6)
>>> [t[i] for i in range(Integer(0), Integer(15))]
[1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 6, 13]

We look at mixing the sparse and the dense:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = L(lambda n: 1, valuation=0); f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: g = LS(lambda n: 1, valuation=1); g
y + y^2 + y^3 + y^4 + y^5 + y^6 + y^7 + O(y^8)
sage: f(g)
1 + y + 2*y^2 + 4*y^3 + 8*y^4 + 16*y^5 + 32*y^6 + O(y^7)

sage: f = z^-2 + 1 + z
sage: g = 1/(y*(1-y)); g
y^-1 + 1 + y + O(y^2)
sage: f(g)
y^-1 + 2 + y + 2*y^2 - y^3 + 2*y^4 + y^5 + y^6 + y^7 + O(y^8)
sage: g^-2 + 1 + g == f(g)
True

sage: f = z^-3 + z^-2 + 1
sage: g = 1/(y^2*(1-y)); g
y^-2 + y^-1 + 1 + O(y)
sage: f(g)
1 + y^4 - 2*y^5 + 2*y^6 - 3*y^7 + 3*y^8 - y^9
sage: g^-3 + g^-2 + 1 == f(g)
True
sage: z(y)
y
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = L(lambda n: Integer(1), valuation=Integer(0)); f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
>>> g = LS(lambda n: Integer(1), valuation=Integer(1)); g
y + y^2 + y^3 + y^4 + y^5 + y^6 + y^7 + O(y^8)
>>> f(g)
1 + y + 2*y^2 + 4*y^3 + 8*y^4 + 16*y^5 + 32*y^6 + O(y^7)

>>> f = z**-Integer(2) + Integer(1) + z
>>> g = Integer(1)/(y*(Integer(1)-y)); g
y^-1 + 1 + y + O(y^2)
>>> f(g)
y^-1 + 2 + y + 2*y^2 - y^3 + 2*y^4 + y^5 + y^6 + y^7 + O(y^8)
>>> g**-Integer(2) + Integer(1) + g == f(g)
True

>>> f = z**-Integer(3) + z**-Integer(2) + Integer(1)
>>> g = Integer(1)/(y**Integer(2)*(Integer(1)-y)); g
y^-2 + y^-1 + 1 + O(y)
>>> f(g)
1 + y^4 - 2*y^5 + 2*y^6 - 3*y^7 + 3*y^8 - y^9
>>> g**-Integer(3) + g**-Integer(2) + Integer(1) == f(g)
True
>>> z(y)
y

We look at cases where the composition does not exist. \(g = 0\) and \(\mathrm{val}(f) < 0\):

sage: g = L(0)
sage: f = z^-1 + z^-2
sage: f.valuation() < 0
True
sage: f(g)
Traceback (most recent call last):
...
ZeroDivisionError: the valuation of the series must be nonnegative
>>> from sage.all import *
>>> g = L(Integer(0))
>>> f = z**-Integer(1) + z**-Integer(2)
>>> f.valuation() < Integer(0)
True
>>> f(g)
Traceback (most recent call last):
...
ZeroDivisionError: the valuation of the series must be nonnegative

\(g \neq 0\) and \(\mathrm{val}(g) \leq 0\) and \(f\) has infinitely many nonzero coefficients:

sage: g = z^-1 + z^-2
sage: g.valuation() <= 0
True
sage: f = L(lambda n: n, valuation=0)
sage: f(g)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: f = L(lambda n: n, valuation=1)
sage: f(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series
>>> from sage.all import *
>>> g = z**-Integer(1) + z**-Integer(2)
>>> g.valuation() <= Integer(0)
True
>>> f = L(lambda n: n, valuation=Integer(0))
>>> f(g)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

>>> f = L(lambda n: n, valuation=Integer(1))
>>> f(Integer(1) + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

We compose the exponential with a Dirichlet series:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: e = L(lambda n: 1/factorial(n), 0)
sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: g = D(constant=1)-1
sage: g                                                                     # needs sage.symbolic
1/(2^s) + 1/(3^s) + 1/(4^s) + O(1/(5^s))

sage: e(g)[0:10]
[0, 1, 1, 1, 3/2, 1, 2, 1, 13/6, 3/2]

sage: sum(g^k/factorial(k) for k in range(10))[0:10]
[0, 1, 1, 1, 3/2, 1, 2, 1, 13/6, 3/2]

sage: g = D([0,1,0,1,1,2])
sage: g                                                                     # needs sage.symbolic
1/(2^s) + 1/(4^s) + 1/(5^s) + 2/6^s
sage: e(g)[0:10]
[0, 1, 1, 0, 3/2, 1, 2, 0, 7/6, 0]
sage: sum(g^k/factorial(k) for k in range(10))[0:10]
[0, 1, 1, 0, 3/2, 1, 2, 0, 7/6, 0]

sage: e(D([1,0,1]))
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: e5 = L(e, degree=5)
sage: e5
1 + z + 1/2*z^2 + 1/6*z^3 + 1/24*z^4
sage: e5(g)                                                                 # needs sage.symbolic
1 + 1/(2^s) + 3/2/4^s + 1/(5^s) + 2/6^s + O(1/(8^s))
sage: sum(e5[k] * g^k for k in range(5))                                    # needs sage.symbolic
1 + 1/(2^s) + 3/2/4^s + 1/(5^s) + 2/6^s + O(1/(8^s))
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> e = L(lambda n: Integer(1)/factorial(n), Integer(0))
>>> D = LazyDirichletSeriesRing(QQ, "s")
>>> g = D(constant=Integer(1))-Integer(1)
>>> g                                                                     # needs sage.symbolic
1/(2^s) + 1/(3^s) + 1/(4^s) + O(1/(5^s))

>>> e(g)[Integer(0):Integer(10)]
[0, 1, 1, 1, 3/2, 1, 2, 1, 13/6, 3/2]

>>> sum(g**k/factorial(k) for k in range(Integer(10)))[Integer(0):Integer(10)]
[0, 1, 1, 1, 3/2, 1, 2, 1, 13/6, 3/2]

>>> g = D([Integer(0),Integer(1),Integer(0),Integer(1),Integer(1),Integer(2)])
>>> g                                                                     # needs sage.symbolic
1/(2^s) + 1/(4^s) + 1/(5^s) + 2/6^s
>>> e(g)[Integer(0):Integer(10)]
[0, 1, 1, 0, 3/2, 1, 2, 0, 7/6, 0]
>>> sum(g**k/factorial(k) for k in range(Integer(10)))[Integer(0):Integer(10)]
[0, 1, 1, 0, 3/2, 1, 2, 0, 7/6, 0]

>>> e(D([Integer(1),Integer(0),Integer(1)]))
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

>>> e5 = L(e, degree=Integer(5))
>>> e5
1 + z + 1/2*z^2 + 1/6*z^3 + 1/24*z^4
>>> e5(g)                                                                 # needs sage.symbolic
1 + 1/(2^s) + 3/2/4^s + 1/(5^s) + 2/6^s + O(1/(8^s))
>>> sum(e5[k] * g**k for k in range(Integer(5)))                                    # needs sage.symbolic
1 + 1/(2^s) + 3/2/4^s + 1/(5^s) + 2/6^s + O(1/(8^s))

The output parent is always the common parent between the base ring of \(f\) and the parent of \(g\) or extended to the corresponding lazy series:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: R.<x> = ZZ[]
sage: parent(z(x))
Univariate Polynomial Ring in x over Rational Field
sage: parent(z(R.zero()))
Univariate Polynomial Ring in x over Rational Field
sage: parent(z(0))
Rational Field
sage: f = 1 / (1 - z)
sage: f(x)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + O(x^7)
sage: three = L(3)(x^2); three
3
sage: parent(three)
Univariate Polynomial Ring in x over Rational Field
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> parent(z(x))
Univariate Polynomial Ring in x over Rational Field
>>> parent(z(R.zero()))
Univariate Polynomial Ring in x over Rational Field
>>> parent(z(Integer(0)))
Rational Field
>>> f = Integer(1) / (Integer(1) - z)
>>> f(x)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + O(x^7)
>>> three = L(Integer(3))(x**Integer(2)); three
3
>>> parent(three)
Univariate Polynomial Ring in x over Rational Field

Consistency check when \(g\) is an uninitialized series between a polynomial \(f\) as both a polynomial and a lazy series:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = 1 + z
sage: g = L.undefined(valuation=0)
sage: f(g) == f.polynomial()(g)
True
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = Integer(1) + z
>>> g = L.undefined(valuation=Integer(0))
>>> f(g) == f.polynomial()(g)
True
compositional_inverse()[source]

Return the compositional inverse of self.

Given a Laurent series \(f\), the compositional inverse is a Laurent series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

  • \(\mathrm{val}(f) = 1\), or

  • \(f = a + b z\) with \(a, b \neq 0\), or

  • \(f = a/z\) with \(a \neq 0\).

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: (2*z).revert()
1/2*z
sage: (2/z).revert()
2*z^-1
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

sage: s = L(degree=1, constant=-1)
sage: s.revert()
-z - z^2 - z^3 + O(z^4)

sage: s = L(degree=1, constant=1)
sage: s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> (Integer(2)*z).revert()
1/2*z
>>> (Integer(2)/z).revert()
2*z^-1
>>> (z-z**Integer(2)).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

>>> s = L(degree=Integer(1), constant=-Integer(1))
>>> s.revert()
-z - z^2 - z^3 + O(z^4)

>>> s = L(degree=Integer(1), constant=Integer(1))
>>> s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)

Warning

For series not known to be eventually constant (e.g., being defined by a function) with approximate valuation \(\leq 1\) (but not necessarily its true valuation), this assumes that this is the actual valuation:

sage: f = L(lambda n: n if n > 2 else 0, valuation=1)
sage: f.revert()
<repr(... failed: ValueError: inverse does not exist>
>>> from sage.all import *
>>> f = L(lambda n: n if n > Integer(2) else Integer(0), valuation=Integer(1))
>>> f.revert()
<repr(... failed: ValueError: inverse does not exist>
derivative(*args)[source]

Return the derivative of the Laurent series.

Multiple variables and iteration counts may be supplied; see the documentation of sage.calculus.functional.derivative() function for details.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: z.derivative()
1
sage: (1+z+z^2).derivative(3)
0
sage: (1/z).derivative()
-z^-2
sage: (1/(1-z)).derivative(z)
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> z.derivative()
1
>>> (Integer(1)+z+z**Integer(2)).derivative(Integer(3))
0
>>> (Integer(1)/z).derivative()
-z^-2
>>> (Integer(1)/(Integer(1)-z)).derivative(z)
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)
integral(variable, constants=None)[source]

Return the integral of self with respect to variable.

INPUT:

  • variable – (optional) the variable to integrate

  • constants – (optional; keyword-only) list of integration constants for the integrals of self (the last constant corresponds to the first integral)

If the first argument is a list, then this method iterprets it as integration constants. If it is a positive integer, the method interprets it as the number of times to integrate the function. If variable is not the variable of the Laurent series, then the coefficients are integrated with respect to variable.

If the integration constants are not specified, they are considered to be \(0\).

EXAMPLES:

sage: L.<t> = LazyLaurentSeriesRing(QQ)
sage: f = t^-3 + 2 + 3*t + t^5
sage: f.integral()
-1/2*t^-2 + 2*t + 3/2*t^2 + 1/6*t^6
sage: f.integral([-2, -2])
1/2*t^-1 - 2 - 2*t + t^2 + 1/2*t^3 + 1/42*t^7
sage: f.integral(t)
-1/2*t^-2 + 2*t + 3/2*t^2 + 1/6*t^6
sage: f.integral(2)
1/2*t^-1 + t^2 + 1/2*t^3 + 1/42*t^7
sage: L.zero().integral()
0
sage: L.zero().integral([0, 1, 2, 3])
t + t^2 + 1/2*t^3
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('t',)); (t,) = L._first_ngens(1)
>>> f = t**-Integer(3) + Integer(2) + Integer(3)*t + t**Integer(5)
>>> f.integral()
-1/2*t^-2 + 2*t + 3/2*t^2 + 1/6*t^6
>>> f.integral([-Integer(2), -Integer(2)])
1/2*t^-1 - 2 - 2*t + t^2 + 1/2*t^3 + 1/42*t^7
>>> f.integral(t)
-1/2*t^-2 + 2*t + 3/2*t^2 + 1/6*t^6
>>> f.integral(Integer(2))
1/2*t^-1 + t^2 + 1/2*t^3 + 1/42*t^7
>>> L.zero().integral()
0
>>> L.zero().integral([Integer(0), Integer(1), Integer(2), Integer(3)])
t + t^2 + 1/2*t^3

We solve the ODE \(f' = a f\) by integrating both sides and the recursive definition:

sage: R.<a, C> = QQ[]
sage: L.<x> = LazyLaurentSeriesRing(R)
sage: f = L.undefined(0)
sage: f.define((a*f).integral(constants=[C]))
sage: f
C + a*C*x + 1/2*a^2*C*x^2 + 1/6*a^3*C*x^3 + 1/24*a^4*C*x^4
 + 1/120*a^5*C*x^5 + 1/720*a^6*C*x^6 + O(x^7)
sage: C * exp(a*x)
C + a*C*x + 1/2*a^2*C*x^2 + 1/6*a^3*C*x^3 + 1/24*a^4*C*x^4
 + 1/120*a^5*C*x^5 + 1/720*a^6*C*x^6 + O(x^7)
>>> from sage.all import *
>>> R = QQ['a, C']; (a, C,) = R._first_ngens(2)
>>> L = LazyLaurentSeriesRing(R, names=('x',)); (x,) = L._first_ngens(1)
>>> f = L.undefined(Integer(0))
>>> f.define((a*f).integral(constants=[C]))
>>> f
C + a*C*x + 1/2*a^2*C*x^2 + 1/6*a^3*C*x^3 + 1/24*a^4*C*x^4
 + 1/120*a^5*C*x^5 + 1/720*a^6*C*x^6 + O(x^7)
>>> C * exp(a*x)
C + a*C*x + 1/2*a^2*C*x^2 + 1/6*a^3*C*x^3 + 1/24*a^4*C*x^4
 + 1/120*a^5*C*x^5 + 1/720*a^6*C*x^6 + O(x^7)

We can integrate both the series and coefficients:

sage: R.<x,y,z> = QQ[]
sage: L.<t> = LazyLaurentSeriesRing(R)
sage: f = (x*t^2 + y*t^-2 + z)^2; f
y^2*t^-4 + 2*y*z*t^-2 + (2*x*y + z^2) + 2*x*z*t^2 + x^2*t^4
sage: f.integral(x)
x*y^2*t^-4 + 2*x*y*z*t^-2 + (x^2*y + x*z^2) + x^2*z*t^2 + 1/3*x^3*t^4
sage: f.integral(t)
-1/3*y^2*t^-3 - 2*y*z*t^-1 + (2*x*y + z^2)*t + 2/3*x*z*t^3 + 1/5*x^2*t^5
sage: f.integral(y, constants=[x*y*z])
-1/9*y^3*t^-3 - y^2*z*t^-1 + x*y*z + (x*y^2 + y*z^2)*t + 2/3*x*y*z*t^3 + 1/5*x^2*y*t^5
>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> L = LazyLaurentSeriesRing(R, names=('t',)); (t,) = L._first_ngens(1)
>>> f = (x*t**Integer(2) + y*t**-Integer(2) + z)**Integer(2); f
y^2*t^-4 + 2*y*z*t^-2 + (2*x*y + z^2) + 2*x*z*t^2 + x^2*t^4
>>> f.integral(x)
x*y^2*t^-4 + 2*x*y*z*t^-2 + (x^2*y + x*z^2) + x^2*z*t^2 + 1/3*x^3*t^4
>>> f.integral(t)
-1/3*y^2*t^-3 - 2*y*z*t^-1 + (2*x*y + z^2)*t + 2/3*x*z*t^3 + 1/5*x^2*t^5
>>> f.integral(y, constants=[x*y*z])
-1/9*y^3*t^-3 - y^2*z*t^-1 + x*y*z + (x*y^2 + y*z^2)*t + 2/3*x*y*z*t^3 + 1/5*x^2*y*t^5
is_unit()[source]

Return whether this element is a unit in the ring.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: (2*z).is_unit()
False

sage: (1 + 2*z).is_unit()
True

sage: (1 + 2*z^-1).is_unit()
False

sage: (z^3 + 4 - z^-2).is_unit()
True
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> (Integer(2)*z).is_unit()
False

>>> (Integer(1) + Integer(2)*z).is_unit()
True

>>> (Integer(1) + Integer(2)*z**-Integer(1)).is_unit()
False

>>> (z**Integer(3) + Integer(4) - z**-Integer(2)).is_unit()
True
polynomial(degree=None, name=None)[source]

Return self as a Laurent polynomial if self is actually so.

INPUT:

  • degreeNone or an integer

  • name – name of the variable; if it is None, the name of the variable of the series is used

OUTPUT:

A Laurent polynomial if the valuation of the series is negative or a polynomial otherwise.

If degree is not None, the terms of the series of degree greater than degree are first truncated. If degree is None and the series is not a polynomial or a Laurent polynomial, a ValueError is raised.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = L([1,0,0,2,0,0,0,3], valuation=5); f
z^5 + 2*z^8 + 3*z^12
sage: f.polynomial()
3*z^12 + 2*z^8 + z^5
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = L([Integer(1),Integer(0),Integer(0),Integer(2),Integer(0),Integer(0),Integer(0),Integer(3)], valuation=Integer(5)); f
z^5 + 2*z^8 + 3*z^12
>>> f.polynomial()
3*z^12 + 2*z^8 + z^5
revert()[source]

Return the compositional inverse of self.

Given a Laurent series \(f\), the compositional inverse is a Laurent series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

  • \(\mathrm{val}(f) = 1\), or

  • \(f = a + b z\) with \(a, b \neq 0\), or

  • \(f = a/z\) with \(a \neq 0\).

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: (2*z).revert()
1/2*z
sage: (2/z).revert()
2*z^-1
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

sage: s = L(degree=1, constant=-1)
sage: s.revert()
-z - z^2 - z^3 + O(z^4)

sage: s = L(degree=1, constant=1)
sage: s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> (Integer(2)*z).revert()
1/2*z
>>> (Integer(2)/z).revert()
2*z^-1
>>> (z-z**Integer(2)).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

>>> s = L(degree=Integer(1), constant=-Integer(1))
>>> s.revert()
-z - z^2 - z^3 + O(z^4)

>>> s = L(degree=Integer(1), constant=Integer(1))
>>> s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)

Warning

For series not known to be eventually constant (e.g., being defined by a function) with approximate valuation \(\leq 1\) (but not necessarily its true valuation), this assumes that this is the actual valuation:

sage: f = L(lambda n: n if n > 2 else 0, valuation=1)
sage: f.revert()
<repr(... failed: ValueError: inverse does not exist>
>>> from sage.all import *
>>> f = L(lambda n: n if n > Integer(2) else Integer(0), valuation=Integer(1))
>>> f.revert()
<repr(... failed: ValueError: inverse does not exist>
class sage.rings.lazy_series.LazyModuleElement(parent, coeff_stream)[source]

Bases: Element

A lazy sequence with a module structure given by term-wise addition and scalar multiplication.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: M = L(lambda n: n, valuation=0)
sage: N = L(lambda n: 1, valuation=0)
sage: M[0:10]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: N[0:10]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> M = L(lambda n: n, valuation=Integer(0))
>>> N = L(lambda n: Integer(1), valuation=Integer(0))
>>> M[Integer(0):Integer(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> N[Integer(0):Integer(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Two sequences can be added:

sage: O = M + N
sage: O[0:10]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
>>> from sage.all import *
>>> O = M + N
>>> O[Integer(0):Integer(10)]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Two sequences can be subtracted:

sage: P = M - N
sage: P[0:10]
[-1, 0, 1, 2, 3, 4, 5, 6, 7, 8]
>>> from sage.all import *
>>> P = M - N
>>> P[Integer(0):Integer(10)]
[-1, 0, 1, 2, 3, 4, 5, 6, 7, 8]

A sequence can be multiplied by a scalar:

sage: Q = 2 * M
sage: Q[0:10]
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]
>>> from sage.all import *
>>> Q = Integer(2) * M
>>> Q[Integer(0):Integer(10)]
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

The negation of a sequence can also be found:

sage: R = -M
sage: R[0:10]
[0, -1, -2, -3, -4, -5, -6, -7, -8, -9]
>>> from sage.all import *
>>> R = -M
>>> R[Integer(0):Integer(10)]
[0, -1, -2, -3, -4, -5, -6, -7, -8, -9]
arccos()[source]

Return the arccosine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(RR)
sage: arccos(z)                                                             # needs sage.symbolic
1.57079632679490 - 1.00000000000000*z + 0.000000000000000*z^2
 - 0.166666666666667*z^3 + 0.000000000000000*z^4
 - 0.0750000000000000*z^5 + O(1.00000000000000*z^7)

sage: L.<z> = LazyLaurentSeriesRing(SR)                                     # needs sage.symbolic
sage: arccos(z/(1-z))                                                       # needs sage.symbolic
1/2*pi - z - z^2 - 7/6*z^3 - 3/2*z^4 - 83/40*z^5 - 73/24*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(SR)                                     # needs sage.symbolic
sage: arccos(x/(1-y))                                                       # needs sage.symbolic
1/2*pi + (-x) + (-x*y) + ((-1/6)*x^3-x*y^2) + ((-1/2)*x^3*y-x*y^3)
 + ((-3/40)*x^5-x^3*y^2-x*y^4) + ((-3/8)*x^5*y+(-5/3)*x^3*y^3-x*y^5) + O(x,y)^7
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(RR, names=('z',)); (z,) = L._first_ngens(1)
>>> arccos(z)                                                             # needs sage.symbolic
1.57079632679490 - 1.00000000000000*z + 0.000000000000000*z^2
 - 0.166666666666667*z^3 + 0.000000000000000*z^4
 - 0.0750000000000000*z^5 + O(1.00000000000000*z^7)

>>> L = LazyLaurentSeriesRing(SR, names=('z',)); (z,) = L._first_ngens(1)# needs sage.symbolic
>>> arccos(z/(Integer(1)-z))                                                       # needs sage.symbolic
1/2*pi - z - z^2 - 7/6*z^3 - 3/2*z^4 - 83/40*z^5 - 73/24*z^6 + O(z^7)

>>> L = LazyPowerSeriesRing(SR, names=('x', 'y',)); (x, y,) = L._first_ngens(2)# needs sage.symbolic
>>> arccos(x/(Integer(1)-y))                                                       # needs sage.symbolic
1/2*pi + (-x) + (-x*y) + ((-1/6)*x^3-x*y^2) + ((-1/2)*x^3*y-x*y^3)
 + ((-3/40)*x^5-x^3*y^2-x*y^4) + ((-3/8)*x^5*y+(-5/3)*x^3*y^3-x*y^5) + O(x,y)^7
arccot()[source]

Return the arctangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(RR)
sage: arccot(z)                                                             # needs sage.symbolic
1.57079632679490 - 1.00000000000000*z + 0.000000000000000*z^2
 + 0.333333333333333*z^3 + 0.000000000000000*z^4
 - 0.200000000000000*z^5 + O(1.00000000000000*z^7)

sage: L.<z> = LazyLaurentSeriesRing(SR)                                     # needs sage.symbolic
sage: arccot(z/(1-z))                                                       # needs sage.symbolic
1/2*pi - z - z^2 - 2/3*z^3 + 4/5*z^5 + 4/3*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(SR)                                     # needs sage.symbolic
sage: acot(x/(1-y))                                                         # needs sage.symbolic
1/2*pi + (-x) + (-x*y) + (1/3*x^3-x*y^2) + (x^3*y-x*y^3)
 + ((-1/5)*x^5+2*x^3*y^2-x*y^4) + (-x^5*y+10/3*x^3*y^3-x*y^5) + O(x,y)^7
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(RR, names=('z',)); (z,) = L._first_ngens(1)
>>> arccot(z)                                                             # needs sage.symbolic
1.57079632679490 - 1.00000000000000*z + 0.000000000000000*z^2
 + 0.333333333333333*z^3 + 0.000000000000000*z^4
 - 0.200000000000000*z^5 + O(1.00000000000000*z^7)

>>> L = LazyLaurentSeriesRing(SR, names=('z',)); (z,) = L._first_ngens(1)# needs sage.symbolic
>>> arccot(z/(Integer(1)-z))                                                       # needs sage.symbolic
1/2*pi - z - z^2 - 2/3*z^3 + 4/5*z^5 + 4/3*z^6 + O(z^7)

>>> L = LazyPowerSeriesRing(SR, names=('x', 'y',)); (x, y,) = L._first_ngens(2)# needs sage.symbolic
>>> acot(x/(Integer(1)-y))                                                         # needs sage.symbolic
1/2*pi + (-x) + (-x*y) + (1/3*x^3-x*y^2) + (x^3*y-x*y^3)
 + ((-1/5)*x^5+2*x^3*y^2-x*y^4) + (-x^5*y+10/3*x^3*y^3-x*y^5) + O(x,y)^7
arcsin()[source]

Return the arcsine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: arcsin(z)
z + 1/6*z^3 + 3/40*z^5 + 5/112*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: asin(x/(1-y))
x + x*y + (1/6*x^3+x*y^2) + (1/2*x^3*y+x*y^3)
 + (3/40*x^5+x^3*y^2+x*y^4) + (3/8*x^5*y+5/3*x^3*y^3+x*y^5)
 + (5/112*x^7+9/8*x^5*y^2+5/2*x^3*y^4+x*y^6) + O(x,y)^8
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> arcsin(z)
z + 1/6*z^3 + 3/40*z^5 + 5/112*z^7 + O(z^8)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> asin(x/(Integer(1)-y))
x + x*y + (1/6*x^3+x*y^2) + (1/2*x^3*y+x*y^3)
 + (3/40*x^5+x^3*y^2+x*y^4) + (3/8*x^5*y+5/3*x^3*y^3+x*y^5)
 + (5/112*x^7+9/8*x^5*y^2+5/2*x^3*y^4+x*y^6) + O(x,y)^8
arcsinh()[source]

Return the inverse of the hyperbolic sine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: asinh(z)
z - 1/6*z^3 + 3/40*z^5 - 5/112*z^7 + O(z^8)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> asinh(z)
z - 1/6*z^3 + 3/40*z^5 - 5/112*z^7 + O(z^8)

arcsinh is an alias:

sage: arcsinh(z)
z - 1/6*z^3 + 3/40*z^5 - 5/112*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: asinh(x/(1-y))
x + x*y + (-1/6*x^3+x*y^2) + (-1/2*x^3*y+x*y^3) + (3/40*x^5-x^3*y^2+x*y^4)
 + (3/8*x^5*y-5/3*x^3*y^3+x*y^5) + (-5/112*x^7+9/8*x^5*y^2-5/2*x^3*y^4+x*y^6) + O(x,y)^8
>>> from sage.all import *
>>> arcsinh(z)
z - 1/6*z^3 + 3/40*z^5 - 5/112*z^7 + O(z^8)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> asinh(x/(Integer(1)-y))
x + x*y + (-1/6*x^3+x*y^2) + (-1/2*x^3*y+x*y^3) + (3/40*x^5-x^3*y^2+x*y^4)
 + (3/8*x^5*y-5/3*x^3*y^3+x*y^5) + (-5/112*x^7+9/8*x^5*y^2-5/2*x^3*y^4+x*y^6) + O(x,y)^8
arctan()[source]

Return the arctangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: arctan(z)
z - 1/3*z^3 + 1/5*z^5 - 1/7*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: atan(x/(1-y))
x + x*y + (-1/3*x^3+x*y^2) + (-x^3*y+x*y^3) + (1/5*x^5-2*x^3*y^2+x*y^4)
 + (x^5*y-10/3*x^3*y^3+x*y^5) + (-1/7*x^7+3*x^5*y^2-5*x^3*y^4+x*y^6) + O(x,y)^8
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> arctan(z)
z - 1/3*z^3 + 1/5*z^5 - 1/7*z^7 + O(z^8)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> atan(x/(Integer(1)-y))
x + x*y + (-1/3*x^3+x*y^2) + (-x^3*y+x*y^3) + (1/5*x^5-2*x^3*y^2+x*y^4)
 + (x^5*y-10/3*x^3*y^3+x*y^5) + (-1/7*x^7+3*x^5*y^2-5*x^3*y^4+x*y^6) + O(x,y)^8
arctanh()[source]

Return the inverse of the hyperbolic tangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: atanh(z)
z + 1/3*z^3 + 1/5*z^5 + 1/7*z^7 + O(z^8)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> atanh(z)
z + 1/3*z^3 + 1/5*z^5 + 1/7*z^7 + O(z^8)

arctanh is an alias:

sage: arctanh(z)
z + 1/3*z^3 + 1/5*z^5 + 1/7*z^7 + O(z^8)

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: atanh(x/(1-y))
x + x*y + (1/3*x^3+x*y^2) + (x^3*y+x*y^3) + (1/5*x^5+2*x^3*y^2+x*y^4)
 + (x^5*y+10/3*x^3*y^3+x*y^5) + (1/7*x^7+3*x^5*y^2+5*x^3*y^4+x*y^6) + O(x,y)^8
>>> from sage.all import *
>>> arctanh(z)
z + 1/3*z^3 + 1/5*z^5 + 1/7*z^7 + O(z^8)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> atanh(x/(Integer(1)-y))
x + x*y + (1/3*x^3+x*y^2) + (x^3*y+x*y^3) + (1/5*x^5+2*x^3*y^2+x*y^4)
 + (x^5*y+10/3*x^3*y^3+x*y^5) + (1/7*x^7+3*x^5*y^2+5*x^3*y^4+x*y^6) + O(x,y)^8
change_ring(ring)[source]

Return self with coefficients converted to elements of ring.

INPUT:

  • ring – a ring

EXAMPLES:

Dense Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: s = 2 + z
sage: t = s.change_ring(QQ)
sage: t^-1
1/2 - 1/4*z + 1/8*z^2 - 1/16*z^3 + 1/32*z^4 - 1/64*z^5 + 1/128*z^6 + O(z^7)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: N = M.change_ring(QQ)
sage: N.parent()
Lazy Laurent Series Ring in z over Rational Field
sage: M.parent()
Lazy Laurent Series Ring in z over Integer Ring
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, sparse=False, names=('z',)); (z,) = L._first_ngens(1)
>>> s = Integer(2) + z
>>> t = s.change_ring(QQ)
>>> t**-Integer(1)
1/2 - 1/4*z + 1/8*z^2 - 1/16*z^3 + 1/32*z^4 - 1/64*z^5 + 1/128*z^6 + O(z^7)
>>> M = L(lambda n: n, valuation=Integer(0)); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
>>> N = M.change_ring(QQ)
>>> N.parent()
Lazy Laurent Series Ring in z over Rational Field
>>> M.parent()
Lazy Laurent Series Ring in z over Integer Ring

Sparse Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.parent()
Lazy Laurent Series Ring in z over Integer Ring
sage: N = M.change_ring(QQ)
sage: N.parent()
Lazy Laurent Series Ring in z over Rational Field
sage: M^-1
z^-1 - 2 + z + O(z^6)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, sparse=True, names=('z',)); (z,) = L._first_ngens(1)
>>> M = L(lambda n: n, valuation=Integer(0)); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
>>> M.parent()
Lazy Laurent Series Ring in z over Integer Ring
>>> N = M.change_ring(QQ)
>>> N.parent()
Lazy Laurent Series Ring in z over Rational Field
>>> M**-Integer(1)
z^-1 - 2 + z + O(z^6)

A Dirichlet series example:

sage: L = LazyDirichletSeriesRing(ZZ, 'z')
sage: s = L(constant=2)
sage: t = s.change_ring(QQ)
sage: t.parent()
Lazy Dirichlet Series Ring in z over Rational Field
sage: it = t^-1
sage: it                                                                    # needs sage.symbolic
1/2 - 1/2/2^z - 1/2/3^z - 1/2/5^z + 1/2/6^z - 1/2/7^z + O(1/(8^z))
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, 'z')
>>> s = L(constant=Integer(2))
>>> t = s.change_ring(QQ)
>>> t.parent()
Lazy Dirichlet Series Ring in z over Rational Field
>>> it = t**-Integer(1)
>>> it                                                                    # needs sage.symbolic
1/2 - 1/2/2^z - 1/2/3^z - 1/2/5^z + 1/2/6^z - 1/2/7^z + O(1/(8^z))

A Taylor series example:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: s = 2 + z
sage: t = s.change_ring(QQ)
sage: t^-1
1/2 - 1/4*z + 1/8*z^2 - 1/16*z^3 + 1/32*z^4 - 1/64*z^5 + 1/128*z^6 + O(z^7)
sage: t.parent()
Lazy Taylor Series Ring in z over Rational Field
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> s = Integer(2) + z
>>> t = s.change_ring(QQ)
>>> t**-Integer(1)
1/2 - 1/4*z + 1/8*z^2 - 1/16*z^3 + 1/32*z^4 - 1/64*z^5 + 1/128*z^6 + O(z^7)
>>> t.parent()
Lazy Taylor Series Ring in z over Rational Field
coefficient(n)[source]

Return the homogeneous degree n part of the series.

INPUT:

  • n – integer; the degree

For a series f, the slice f[start:stop] produces the following:

  • if start and stop are integers, return the list of terms with given degrees

  • if start is None, return the list of terms beginning with the valuation

  • if stop is None, return a lazy_list_generic instead.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = z / (1 - 2*z^3)
sage: [f[n] for n in range(20)]
[0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
sage: f[0:20]
[0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
sage: f[:20]
[1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
sage: f[::3]
lazy list [1, 2, 4, ...]

sage: M = L(lambda n: n, valuation=0)
sage: [M[n] for n in range(20)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0)
sage: [M[n] for n in range(20)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = z / (Integer(1) - Integer(2)*z**Integer(3))
>>> [f[n] for n in range(Integer(20))]
[0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
>>> f[Integer(0):Integer(20)]
[0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
>>> f[:Integer(20)]
[1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
>>> f[::Integer(3)]
lazy list [1, 2, 4, ...]

>>> M = L(lambda n: n, valuation=Integer(0))
>>> [M[n] for n in range(Integer(20))]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

>>> L = LazyLaurentSeriesRing(ZZ, sparse=True, names=('z',)); (z,) = L._first_ngens(1)
>>> M = L(lambda n: n, valuation=Integer(0))
>>> [M[n] for n in range(Integer(20))]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Similarly for multivariate series:

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sin(x*y)[:11]
[x*y, 0, 0, 0, -1/6*x^3*y^3, 0, 0, 0, 1/120*x^5*y^5]
sage: sin(x*y)[2::4]
lazy list [x*y, -1/6*x^3*y^3, 1/120*x^5*y^5, ...]
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> sin(x*y)[:Integer(11)]
[x*y, 0, 0, 0, -1/6*x^3*y^3, 0, 0, 0, 1/120*x^5*y^5]
>>> sin(x*y)[Integer(2)::Integer(4)]
lazy list [x*y, -1/6*x^3*y^3, 1/120*x^5*y^5, ...]

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: L(lambda n: n)[1:11]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, "z")
>>> L(lambda n: n)[Integer(1):Integer(11)]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
coefficients(n=None)[source]

Return the first \(n\) nonzero coefficients of self.

INPUT:

  • n – (optional) the number of nonzero coefficients to return

If the series has fewer than \(n\) nonzero coefficients, only these are returned.

If n is None, a lazy_list_generic with all nonzero coefficients is returned instead.

Warning

If there are fewer than \(n\) nonzero coefficients, but this cannot be detected, this method will not return.

EXAMPLES:

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: f = L([1,2,3])
sage: f.coefficients(5)
doctest:...: DeprecationWarning: the method coefficients now only returns the nonzero coefficients. Use __getitem__ instead.
See https://github.com/sagemath/sage/issues/32367 for details.
[1, 2, 3]

sage: f = sin(x)
sage: f.coefficients(5)
[1, -1/6, 1/120, -1/5040, 1/362880]

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: f = sin(x^2+y^2)
sage: f.coefficients(5)
[1, 1, -1/6, -1/2, -1/2]

sage: f.coefficients()
lazy list [1, 1, -1/6, ...]

sage: L.<x> = LazyPowerSeriesRing(GF(2))
sage: f = L(lambda n: n)
sage: f.coefficients(5)
[1, 1, 1, 1, 1]
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x',)); (x,) = L._first_ngens(1)
>>> f = L([Integer(1),Integer(2),Integer(3)])
>>> f.coefficients(Integer(5))
doctest:...: DeprecationWarning: the method coefficients now only returns the nonzero coefficients. Use __getitem__ instead.
See https://github.com/sagemath/sage/issues/32367 for details.
[1, 2, 3]

>>> f = sin(x)
>>> f.coefficients(Integer(5))
[1, -1/6, 1/120, -1/5040, 1/362880]

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> f = sin(x**Integer(2)+y**Integer(2))
>>> f.coefficients(Integer(5))
[1, 1, -1/6, -1/2, -1/2]

>>> f.coefficients()
lazy list [1, 1, -1/6, ...]

>>> L = LazyPowerSeriesRing(GF(Integer(2)), names=('x',)); (x,) = L._first_ngens(1)
>>> f = L(lambda n: n)
>>> f.coefficients(Integer(5))
[1, 1, 1, 1, 1]
cos()[source]

Return the cosine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cos(z)
1 - 1/2*z^2 + 1/24*z^4 - 1/720*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: cos(x/(1-y)).polynomial(4)
1/24*x^4 - 3/2*x^2*y^2 - x^2*y - 1/2*x^2 + 1
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> cos(z)
1 - 1/2*z^2 + 1/24*z^4 - 1/720*z^6 + O(z^7)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> cos(x/(Integer(1)-y)).polynomial(Integer(4))
1/24*x^4 - 3/2*x^2*y^2 - x^2*y - 1/2*x^2 + 1
cosh()[source]

Return the hyperbolic cosine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cosh(z)
1 + 1/2*z^2 + 1/24*z^4 + 1/720*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: cosh(x/(1-y))
1 + 1/2*x^2 + x^2*y + (1/24*x^4+3/2*x^2*y^2) + (1/6*x^4*y+2*x^2*y^3)
 + (1/720*x^6+5/12*x^4*y^2+5/2*x^2*y^4) + O(x,y)^7
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> cosh(z)
1 + 1/2*z^2 + 1/24*z^4 + 1/720*z^6 + O(z^7)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> cosh(x/(Integer(1)-y))
1 + 1/2*x^2 + x^2*y + (1/24*x^4+3/2*x^2*y^2) + (1/6*x^4*y+2*x^2*y^3)
 + (1/720*x^6+5/12*x^4*y^2+5/2*x^2*y^4) + O(x,y)^7
cot()[source]

Return the cotangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cot(z)
z^-1 - 1/3*z - 1/45*z^3 - 2/945*z^5 + O(z^6)

sage: L.<x> = LazyLaurentSeriesRing(QQ)
sage: cot(x/(1-x)).polynomial(4)
x^-1 - 1 - 1/3*x - 1/3*x^2 - 16/45*x^3 - 2/5*x^4
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> cot(z)
z^-1 - 1/3*z - 1/45*z^3 - 2/945*z^5 + O(z^6)

>>> L = LazyLaurentSeriesRing(QQ, names=('x',)); (x,) = L._first_ngens(1)
>>> cot(x/(Integer(1)-x)).polynomial(Integer(4))
x^-1 - 1 - 1/3*x - 1/3*x^2 - 16/45*x^3 - 2/5*x^4
coth()[source]

Return the hyperbolic cotangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: coth(z)                                                               # needs sage.libs.flint
z^-1 + 1/3*z - 1/45*z^3 + 2/945*z^5 + O(z^6)

sage: coth(z + z^2)                                                         # needs sage.libs.flint
z^-1 - 1 + 4/3*z - 2/3*z^2 + 44/45*z^3 - 16/15*z^4 + 884/945*z^5 + O(z^6)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> coth(z)                                                               # needs sage.libs.flint
z^-1 + 1/3*z - 1/45*z^3 + 2/945*z^5 + O(z^6)

>>> coth(z + z**Integer(2))                                                         # needs sage.libs.flint
z^-1 - 1 + 4/3*z - 2/3*z^2 + 44/45*z^3 - 16/15*z^4 + 884/945*z^5 + O(z^6)
csc()[source]

Return the cosecant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csc(z)
z^-1 + 1/6*z + 7/360*z^3 + 31/15120*z^5 + O(z^6)

sage: L.<x> = LazyLaurentSeriesRing(QQ)
sage: csc(x/(1-x)).polynomial(4)
x^-1 - 1 + 1/6*x + 1/6*x^2 + 67/360*x^3 + 9/40*x^4
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> csc(z)
z^-1 + 1/6*z + 7/360*z^3 + 31/15120*z^5 + O(z^6)

>>> L = LazyLaurentSeriesRing(QQ, names=('x',)); (x,) = L._first_ngens(1)
>>> csc(x/(Integer(1)-x)).polynomial(Integer(4))
x^-1 - 1 + 1/6*x + 1/6*x^2 + 67/360*x^3 + 9/40*x^4
csch()[source]

Return the hyperbolic cosecant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csch(z)                                                               # needs sage.libs.flint
z^-1 - 1/6*z + 7/360*z^3 - 31/15120*z^5 + O(z^6)

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csch(z/(1-z))                                                         # needs sage.libs.flint
z^-1 - 1 - 1/6*z - 1/6*z^2 - 53/360*z^3 - 13/120*z^4 - 787/15120*z^5 + O(z^6)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> csch(z)                                                               # needs sage.libs.flint
z^-1 - 1/6*z + 7/360*z^3 - 31/15120*z^5 + O(z^6)

>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> csch(z/(Integer(1)-z))                                                         # needs sage.libs.flint
z^-1 - 1 - 1/6*z - 1/6*z^2 - 53/360*z^3 - 13/120*z^4 - 787/15120*z^5 + O(z^6)
define(s)[source]

Define an equation by self = s.

INPUT:

  • s – a lazy series

EXAMPLES:

We begin by constructing the Catalan numbers:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: C = L.undefined()
sage: C.define(1 + z*C^2)
sage: C
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)
sage: binomial(2000, 1000) / C[1000]                                        # needs sage.symbolic
1001
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> C = L.undefined()
>>> C.define(Integer(1) + z*C**Integer(2))
>>> C
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)
>>> binomial(Integer(2000), Integer(1000)) / C[Integer(1000)]                                        # needs sage.symbolic
1001

The Catalan numbers but with a valuation \(1\):

sage: B = L.undefined(valuation=1)
sage: B.define(z + B^2)
sage: B
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)
>>> from sage.all import *
>>> B = L.undefined(valuation=Integer(1))
>>> B.define(z + B**Integer(2))
>>> B
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

We can define multiple series that are linked:

sage: s = L.undefined()
sage: t = L.undefined()
sage: s.define(1 + z*t^3)
sage: t.define(1 + z*s^2)
sage: s[0:9]
[1, 1, 3, 9, 34, 132, 546, 2327, 10191]
sage: t[0:9]
[1, 1, 2, 7, 24, 95, 386, 1641, 7150]
>>> from sage.all import *
>>> s = L.undefined()
>>> t = L.undefined()
>>> s.define(Integer(1) + z*t**Integer(3))
>>> t.define(Integer(1) + z*s**Integer(2))
>>> s[Integer(0):Integer(9)]
[1, 1, 3, 9, 34, 132, 546, 2327, 10191]
>>> t[Integer(0):Integer(9)]
[1, 1, 2, 7, 24, 95, 386, 1641, 7150]

A bigger example:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: A = L.undefined(valuation=5)
sage: B = L.undefined()
sage: C = L.undefined(valuation=2)
sage: A.define(z^5 + B^2)
sage: B.define(z^5 + C^2)
sage: C.define(z^2 + C^2 + A^2)
sage: A[0:15]
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 5, 4, 14, 10, 48]
sage: B[0:15]
[0, 0, 0, 0, 1, 1, 2, 0, 5, 0, 14, 0, 44, 0, 138]
sage: C[0:15]
[0, 0, 1, 0, 1, 0, 2, 0, 5, 0, 15, 0, 44, 2, 142]
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> A = L.undefined(valuation=Integer(5))
>>> B = L.undefined()
>>> C = L.undefined(valuation=Integer(2))
>>> A.define(z**Integer(5) + B**Integer(2))
>>> B.define(z**Integer(5) + C**Integer(2))
>>> C.define(z**Integer(2) + C**Integer(2) + A**Integer(2))
>>> A[Integer(0):Integer(15)]
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 5, 4, 14, 10, 48]
>>> B[Integer(0):Integer(15)]
[0, 0, 0, 0, 1, 1, 2, 0, 5, 0, 14, 0, 44, 0, 138]
>>> C[Integer(0):Integer(15)]
[0, 0, 1, 0, 1, 0, 2, 0, 5, 0, 15, 0, 44, 2, 142]

Counting binary trees:

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: s = L.undefined(valuation=1)
sage: s.define(z + (s^2+s(z^2))/2)
sage: s[0:9]
[0, 1, 1, 1, 2, 3, 6, 11, 23]
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> s = L.undefined(valuation=Integer(1))
>>> s.define(z + (s**Integer(2)+s(z**Integer(2)))/Integer(2))
>>> s[Integer(0):Integer(9)]
[0, 1, 1, 1, 2, 3, 6, 11, 23]

The \(q\)-Catalan numbers:

sage: R.<q> = ZZ[]
sage: L.<z> = LazyLaurentSeriesRing(R)
sage: s = L.undefined(valuation=0)
sage: s.define(1+z*s*s(q*z))
sage: s
1 + z + (q + 1)*z^2 + (q^3 + q^2 + 2*q + 1)*z^3
 + (q^6 + q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 3*q + 1)*z^4
 + (q^10 + q^9 + 2*q^8 + 3*q^7 + 5*q^6 + 5*q^5 + 7*q^4 + 7*q^3 + 6*q^2 + 4*q + 1)*z^5
 + (q^15 + q^14 + 2*q^13 + 3*q^12 + 5*q^11 + 7*q^10 + 9*q^9 + 11*q^8
    + 14*q^7 + 16*q^6 + 16*q^5 + 17*q^4 + 14*q^3 + 10*q^2 + 5*q + 1)*z^6 + O(z^7)
>>> from sage.all import *
>>> R = ZZ['q']; (q,) = R._first_ngens(1)
>>> L = LazyLaurentSeriesRing(R, names=('z',)); (z,) = L._first_ngens(1)
>>> s = L.undefined(valuation=Integer(0))
>>> s.define(Integer(1)+z*s*s(q*z))
>>> s
1 + z + (q + 1)*z^2 + (q^3 + q^2 + 2*q + 1)*z^3
 + (q^6 + q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 3*q + 1)*z^4
 + (q^10 + q^9 + 2*q^8 + 3*q^7 + 5*q^6 + 5*q^5 + 7*q^4 + 7*q^3 + 6*q^2 + 4*q + 1)*z^5
 + (q^15 + q^14 + 2*q^13 + 3*q^12 + 5*q^11 + 7*q^10 + 9*q^9 + 11*q^8
    + 14*q^7 + 16*q^6 + 16*q^5 + 17*q^4 + 14*q^3 + 10*q^2 + 5*q + 1)*z^6 + O(z^7)

We count unlabeled ordered trees by total number of nodes and number of internal nodes:

sage: R.<q> = QQ[]
sage: Q.<z> = LazyPowerSeriesRing(R)
sage: leaf = z
sage: internal_node = q * z
sage: L = Q(constant=1, degree=1)
sage: T = Q.undefined(valuation=1)
sage: T.define(leaf + internal_node * L(T))
sage: T[0:6]
[0, 1, q, q^2 + q, q^3 + 3*q^2 + q, q^4 + 6*q^3 + 6*q^2 + q]
>>> from sage.all import *
>>> R = QQ['q']; (q,) = R._first_ngens(1)
>>> Q = LazyPowerSeriesRing(R, names=('z',)); (z,) = Q._first_ngens(1)
>>> leaf = z
>>> internal_node = q * z
>>> L = Q(constant=Integer(1), degree=Integer(1))
>>> T = Q.undefined(valuation=Integer(1))
>>> T.define(leaf + internal_node * L(T))
>>> T[Integer(0):Integer(6)]
[0, 1, q, q^2 + q, q^3 + 3*q^2 + q, q^4 + 6*q^3 + 6*q^2 + q]

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: g = L(constant=1, valuation=2)
sage: F = L.undefined()
sage: F.define(1 + g*F)
sage: F[:16]
[1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3]
sage: oeis(_)                                                       # optional - internet
0: A002033: Number of perfect partitions of n.
1: A074206: Kalmár's [Kalmar's] problem: number of ordered factorizations of n.
...

sage: F = L.undefined()
sage: F.define(1 + g*F*F)
sage: F[:16]
[1, 1, 1, 3, 1, 5, 1, 10, 3, 5, 1, 24, 1, 5, 5]
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, "z")
>>> g = L(constant=Integer(1), valuation=Integer(2))
>>> F = L.undefined()
>>> F.define(Integer(1) + g*F)
>>> F[:Integer(16)]
[1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3]
>>> oeis(_)                                                       # optional - internet
0: A002033: Number of perfect partitions of n.
1: A074206: Kalmár's [Kalmar's] problem: number of ordered factorizations of n.
...

>>> F = L.undefined()
>>> F.define(Integer(1) + g*F*F)
>>> F[:Integer(16)]
[1, 1, 1, 3, 1, 5, 1, 10, 3, 5, 1, 24, 1, 5, 5]

We can compute the Frobenius character of unlabeled trees:

sage: # needs sage.combinat
sage: m = SymmetricFunctions(QQ).m()
sage: s = SymmetricFunctions(QQ).s()
sage: L = LazySymmetricFunctions(m)
sage: E = L(lambda n: s[n], valuation=0)
sage: X = L(s[1])
sage: A = L.undefined()
sage: A.define(X*E(A))
sage: A[:6]
[m[1],
 2*m[1, 1] + m[2],
 9*m[1, 1, 1] + 5*m[2, 1] + 2*m[3],
 64*m[1, 1, 1, 1] + 34*m[2, 1, 1] + 18*m[2, 2] + 13*m[3, 1] + 4*m[4],
 625*m[1, 1, 1, 1, 1] + 326*m[2, 1, 1, 1] + 171*m[2, 2, 1] + 119*m[3, 1, 1] + 63*m[3, 2] + 35*m[4, 1] + 9*m[5]]
>>> from sage.all import *
>>> # needs sage.combinat
>>> m = SymmetricFunctions(QQ).m()
>>> s = SymmetricFunctions(QQ).s()
>>> L = LazySymmetricFunctions(m)
>>> E = L(lambda n: s[n], valuation=Integer(0))
>>> X = L(s[Integer(1)])
>>> A = L.undefined()
>>> A.define(X*E(A))
>>> A[:Integer(6)]
[m[1],
 2*m[1, 1] + m[2],
 9*m[1, 1, 1] + 5*m[2, 1] + 2*m[3],
 64*m[1, 1, 1, 1] + 34*m[2, 1, 1] + 18*m[2, 2] + 13*m[3, 1] + 4*m[4],
 625*m[1, 1, 1, 1, 1] + 326*m[2, 1, 1, 1] + 171*m[2, 2, 1] + 119*m[3, 1, 1] + 63*m[3, 2] + 35*m[4, 1] + 9*m[5]]
euler()[source]

Return the Euler function evaluated at self.

The Euler function is defined as

\[\phi(z) = (z; z)_{\infty} = \sum_{n=0}^{\infty} (-1)^n q^{(3n^2-n)/2}.\]

EXAMPLES:

sage: L.<q> = LazyLaurentSeriesRing(ZZ)
sage: phi = L.euler()
sage: (q + q^2).euler() - phi(q + q^2)
O(q^7)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('q',)); (q,) = L._first_ngens(1)
>>> phi = L.euler()
>>> (q + q**Integer(2)).euler() - phi(q + q**Integer(2))
O(q^7)
exp()[source]

Return the exponential series of self.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(QQ, "s")
sage: Z = L(constant=1, valuation=2)
sage: exp(Z)                                                                # needs sage.symbolic
1 + 1/(2^s) + 1/(3^s) + 3/2/4^s + 1/(5^s) + 2/6^s + 1/(7^s) + O(1/(8^s))
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(QQ, "s")
>>> Z = L(constant=Integer(1), valuation=Integer(2))
>>> exp(Z)                                                                # needs sage.symbolic
1 + 1/(2^s) + 1/(3^s) + 3/2/4^s + 1/(5^s) + 2/6^s + 1/(7^s) + O(1/(8^s))
hypergeometric(a, b)[source]

Return the \({}_{p}F_{q}\)-hypergeometric function \(\,_pF_{q}\) where \((p,q)\) is the parameterization of self.

INPUT:

  • a – the first parameter of the hypergeometric function

  • b – the second parameter of the hypergeometric function

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: z.hypergeometric([1, 1], [1])
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: z.hypergeometric([], []) - exp(z)
O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: (x+y).hypergeometric([1, 1], [1]).polynomial(4)
x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + y^4 + x^3 + 3*x^2*y
 + 3*x*y^2 + y^3 + x^2 + 2*x*y + y^2 + x + y + 1
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> z.hypergeometric([Integer(1), Integer(1)], [Integer(1)])
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
>>> z.hypergeometric([], []) - exp(z)
O(z^7)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> (x+y).hypergeometric([Integer(1), Integer(1)], [Integer(1)]).polynomial(Integer(4))
x^4 + 4*x^3*y + 6*x^2*y^2 + 4*x*y^3 + y^4 + x^3 + 3*x^2*y
 + 3*x*y^2 + y^3 + x^2 + 2*x*y + y^2 + x + y + 1
is_nonzero(proof=False)[source]

Return True if self is known to be nonzero.

INPUT:

  • proof – boolean (default: False); if True, this will also return an index such that self has a nonzero coefficient

Warning

If the stream is exactly zero, this will run forever.

EXAMPLES:

A series that it not known to be nonzero with no halting precision:

sage: L.<z> = LazyLaurentSeriesRing(GF(2))
sage: f = L(lambda n: 0, valuation=0)
sage: f.is_nonzero()
False
sage: bool(f)
True
sage: g = L(lambda n: 0 if n < 50 else 1, valuation=2)
sage: g.is_nonzero()
False
sage: g[60]
1
sage: g.is_nonzero()
True
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(GF(Integer(2)), names=('z',)); (z,) = L._first_ngens(1)
>>> f = L(lambda n: Integer(0), valuation=Integer(0))
>>> f.is_nonzero()
False
>>> bool(f)
True
>>> g = L(lambda n: Integer(0) if n < Integer(50) else Integer(1), valuation=Integer(2))
>>> g.is_nonzero()
False
>>> g[Integer(60)]
1
>>> g.is_nonzero()
True

With finite halting precision, it can be considered to be indistinguishable from zero until possibly enough coefficients are computed:

sage: L.options.halting_precision = 20
sage: f = L(lambda n: 0, valuation=0)
sage: f.is_zero()
True

sage: g = L(lambda n: 0 if n < 50 else 1, valuation=2)
sage: g.is_nonzero()  # checks up to degree 22 = 2 + 20
False
sage: g.is_nonzero()  # checks up to degree 42 = 22 + 20
False
sage: g.is_nonzero()  # checks up to degree 62 = 42 + 20
True
sage: L.options._reset()
>>> from sage.all import *
>>> L.options.halting_precision = Integer(20)
>>> f = L(lambda n: Integer(0), valuation=Integer(0))
>>> f.is_zero()
True

>>> g = L(lambda n: Integer(0) if n < Integer(50) else Integer(1), valuation=Integer(2))
>>> g.is_nonzero()  # checks up to degree 22 = 2 + 20
False
>>> g.is_nonzero()  # checks up to degree 42 = 22 + 20
False
>>> g.is_nonzero()  # checks up to degree 62 = 42 + 20
True
>>> L.options._reset()

With a proof:

sage: L.<z> = LazyLaurentSeriesRing(GF(5))
sage: g = L(lambda n: 5 if n < 50 else 1, valuation=2)
sage: g.is_nonzero(proof=True)
(True, 50)

sage: L.zero().is_nonzero(proof=True)
(False, None)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(GF(Integer(5)), names=('z',)); (z,) = L._first_ngens(1)
>>> g = L(lambda n: Integer(5) if n < Integer(50) else Integer(1), valuation=Integer(2))
>>> g.is_nonzero(proof=True)
(True, 50)

>>> L.zero().is_nonzero(proof=True)
(False, None)
is_trivial_zero()[source]

Return whether self is known to be trivially zero.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = L(lambda n: 0, valuation=2)
sage: f.is_trivial_zero()
False

sage: L.zero().is_trivial_zero()
True
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = L(lambda n: Integer(0), valuation=Integer(2))
>>> f.is_trivial_zero()
False

>>> L.zero().is_trivial_zero()
True
lift_to_precision(absprec=None)[source]

Return another element of the same parent with absolute precision at least absprec, congruent to this element modulo the precision of this element.

Since the precision of a lazy series is infinity, this method returns the series itself, and the argument is ignored.

EXAMPLES:

sage: P.<t> = PowerSeriesRing(QQ, default_prec=2)
sage: R.<z> = LazyPowerSeriesRing(P)
sage: f = R(lambda n: 1/(1-t)^n)
sage: f
1 + ((1+t+O(t^2))*z) + ((1+2*t+O(t^2))*z^2)
  + ((1+3*t+O(t^2))*z^3)
  + ((1+4*t+O(t^2))*z^4)
  + ((1+5*t+O(t^2))*z^5)
  + ((1+6*t+O(t^2))*z^6) + O(z^7)
sage: f.lift_to_precision()
1 + ((1+t+O(t^2))*z) + ((1+2*t+O(t^2))*z^2)
  + ((1+3*t+O(t^2))*z^3)
  + ((1+4*t+O(t^2))*z^4)
  + ((1+5*t+O(t^2))*z^5)
  + ((1+6*t+O(t^2))*z^6) + O(z^7)
>>> from sage.all import *
>>> P = PowerSeriesRing(QQ, default_prec=Integer(2), names=('t',)); (t,) = P._first_ngens(1)
>>> R = LazyPowerSeriesRing(P, names=('z',)); (z,) = R._first_ngens(1)
>>> f = R(lambda n: Integer(1)/(Integer(1)-t)**n)
>>> f
1 + ((1+t+O(t^2))*z) + ((1+2*t+O(t^2))*z^2)
  + ((1+3*t+O(t^2))*z^3)
  + ((1+4*t+O(t^2))*z^4)
  + ((1+5*t+O(t^2))*z^5)
  + ((1+6*t+O(t^2))*z^6) + O(z^7)
>>> f.lift_to_precision()
1 + ((1+t+O(t^2))*z) + ((1+2*t+O(t^2))*z^2)
  + ((1+3*t+O(t^2))*z^3)
  + ((1+4*t+O(t^2))*z^4)
  + ((1+5*t+O(t^2))*z^5)
  + ((1+6*t+O(t^2))*z^6) + O(z^7)
log()[source]

Return the series for the natural logarithm of self.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(QQ, "s")
sage: Z = L(constant=1)
sage: log(Z)                                                                # needs sage.symbolic
1/(2^s) + 1/(3^s) + 1/2/4^s + 1/(5^s) + 1/(7^s) + O(1/(8^s))
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(QQ, "s")
>>> Z = L(constant=Integer(1))
>>> log(Z)                                                                # needs sage.symbolic
1/(2^s) + 1/(3^s) + 1/2/4^s + 1/(5^s) + 1/(7^s) + O(1/(8^s))
map_coefficients(f)[source]

Return the series with f applied to each nonzero coefficient of self.

INPUT:

  • func – function that takes in a coefficient and returns a new coefficient

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: m = L(lambda n: n, valuation=0); m
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: m.map_coefficients(lambda c: c + 1)
2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + 8*z^7 + O(z^8)
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> m = L(lambda n: n, valuation=Integer(0)); m
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
>>> m.map_coefficients(lambda c: c + Integer(1))
2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + 8*z^7 + O(z^8)

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: s = L(lambda n: n-1)
sage: s                                                                     # needs sage.symbolic
1/(2^z) + 2/3^z + 3/4^z + 4/5^z + 5/6^z + 6/7^z + O(1/(8^z))
sage: ms = s.map_coefficients(lambda c: c + 1)                              # needs sage.symbolic
sage: ms                                                                    # needs sage.symbolic
2/2^z + 3/3^z + 4/4^z + 5/5^z + 6/6^z + 7/7^z + 8/8^z + O(1/(9^z))
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, "z")
>>> s = L(lambda n: n-Integer(1))
>>> s                                                                     # needs sage.symbolic
1/(2^z) + 2/3^z + 3/4^z + 4/5^z + 5/6^z + 6/7^z + O(1/(8^z))
>>> ms = s.map_coefficients(lambda c: c + Integer(1))                              # needs sage.symbolic
>>> ms                                                                    # needs sage.symbolic
2/2^z + 3/3^z + 4/4^z + 5/5^z + 6/6^z + 7/7^z + 8/8^z + O(1/(9^z))

Similarly for multivariate power series:

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: f = 1/(1-(x+y)); f
1 + (x+y) + (x^2+2*x*y+y^2) + (x^3+3*x^2*y+3*x*y^2+y^3)
 + (x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+y^4)
 + (x^5+5*x^4*y+10*x^3*y^2+10*x^2*y^3+5*x*y^4+y^5)
 + (x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5+y^6)
 + O(x,y)^7
sage: f.map_coefficients(lambda c: c^2)
1 + (x+y) + (x^2+4*x*y+y^2) + (x^3+9*x^2*y+9*x*y^2+y^3)
 + (x^4+16*x^3*y+36*x^2*y^2+16*x*y^3+y^4)
 + (x^5+25*x^4*y+100*x^3*y^2+100*x^2*y^3+25*x*y^4+y^5)
 + (x^6+36*x^5*y+225*x^4*y^2+400*x^3*y^3+225*x^2*y^4+36*x*y^5+y^6)
 + O(x,y)^7
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> f = Integer(1)/(Integer(1)-(x+y)); f
1 + (x+y) + (x^2+2*x*y+y^2) + (x^3+3*x^2*y+3*x*y^2+y^3)
 + (x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+y^4)
 + (x^5+5*x^4*y+10*x^3*y^2+10*x^2*y^3+5*x*y^4+y^5)
 + (x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5+y^6)
 + O(x,y)^7
>>> f.map_coefficients(lambda c: c**Integer(2))
1 + (x+y) + (x^2+4*x*y+y^2) + (x^3+9*x^2*y+9*x*y^2+y^3)
 + (x^4+16*x^3*y+36*x^2*y^2+16*x*y^3+y^4)
 + (x^5+25*x^4*y+100*x^3*y^2+100*x^2*y^3+25*x*y^4+y^5)
 + (x^6+36*x^5*y+225*x^4*y^2+400*x^3*y^3+225*x^2*y^4+36*x*y^5+y^6)
 + O(x,y)^7

Similarly for lazy symmetric functions:

sage: # needs sage.combinat
sage: p = SymmetricFunctions(QQ).p()
sage: L = LazySymmetricFunctions(p)
sage: f = 1/(1-2*L(p[1])); f
p[] + 2*p[1] + (4*p[1,1]) + (8*p[1,1,1]) + (16*p[1,1,1,1])
 + (32*p[1,1,1,1,1]) + (64*p[1,1,1,1,1,1]) + O^7
sage: f.map_coefficients(lambda c: log(c, 2))
p[1] + (2*p[1,1]) + (3*p[1,1,1]) + (4*p[1,1,1,1])
 + (5*p[1,1,1,1,1]) + (6*p[1,1,1,1,1,1]) + O^7
>>> from sage.all import *
>>> # needs sage.combinat
>>> p = SymmetricFunctions(QQ).p()
>>> L = LazySymmetricFunctions(p)
>>> f = Integer(1)/(Integer(1)-Integer(2)*L(p[Integer(1)])); f
p[] + 2*p[1] + (4*p[1,1]) + (8*p[1,1,1]) + (16*p[1,1,1,1])
 + (32*p[1,1,1,1,1]) + (64*p[1,1,1,1,1,1]) + O^7
>>> f.map_coefficients(lambda c: log(c, Integer(2)))
p[1] + (2*p[1,1]) + (3*p[1,1,1]) + (4*p[1,1,1,1])
 + (5*p[1,1,1,1,1]) + (6*p[1,1,1,1,1,1]) + O^7
prec()[source]

Return the precision of the series, which is infinity.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1/(1 - z)
sage: f.prec()
+Infinity
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = Integer(1)/(Integer(1) - z)
>>> f.prec()
+Infinity
q_pochhammer(q=None)[source]

Return the infinite q-Pochhammer symbol \((a; q)_{\infty}\), where \(a\) is self.

This is also one version of the quantum dilogarithm or the \(q\)-Exponential function.

INPUT:

  • q – (default: \(q \in \QQ(q)\)) the parameter \(q\)

EXAMPLES:

sage: q = ZZ['q'].fraction_field().gen()
sage: L.<z> = LazyLaurentSeriesRing(q.parent())
sage: qp = L.q_pochhammer(q)
sage: (z + z^2).q_pochhammer(q) - qp(z + z^2)
O(z^7)
>>> from sage.all import *
>>> q = ZZ['q'].fraction_field().gen()
>>> L = LazyLaurentSeriesRing(q.parent(), names=('z',)); (z,) = L._first_ngens(1)
>>> qp = L.q_pochhammer(q)
>>> (z + z**Integer(2)).q_pochhammer(q) - qp(z + z**Integer(2))
O(z^7)
sec()[source]

Return the secant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sec(z)
1 + 1/2*z^2 + 5/24*z^4 + 61/720*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sec(x/(1-y)).polynomial(4)
5/24*x^4 + 3/2*x^2*y^2 + x^2*y + 1/2*x^2 + 1
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> sec(z)
1 + 1/2*z^2 + 5/24*z^4 + 61/720*z^6 + O(z^7)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> sec(x/(Integer(1)-y)).polynomial(Integer(4))
5/24*x^4 + 3/2*x^2*y^2 + x^2*y + 1/2*x^2 + 1
sech()[source]

Return the hyperbolic secant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sech(z)                                                               # needs sage.libs.flint
1 - 1/2*z^2 + 5/24*z^4 - 61/720*z^6 + O(z^7)

sage: L.<x, y> = LazyPowerSeriesRing(QQ)
sage: sech(x/(1-y))                                                         # needs sage.libs.flint
1 + (-1/2*x^2) + (-x^2*y) + (5/24*x^4-3/2*x^2*y^2) + (5/6*x^4*y-2*x^2*y^3)
 + (-61/720*x^6+25/12*x^4*y^2-5/2*x^2*y^4) + O(x,y)^7
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> sech(z)                                                               # needs sage.libs.flint
1 - 1/2*z^2 + 5/24*z^4 - 61/720*z^6 + O(z^7)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> sech(x/(Integer(1)-y))                                                         # needs sage.libs.flint
1 + (-1/2*x^2) + (-x^2*y) + (5/24*x^4-3/2*x^2*y^2) + (5/6*x^4*y-2*x^2*y^3)
 + (-61/720*x^6+25/12*x^4*y^2-5/2*x^2*y^4) + O(x,y)^7
set(s)[source]

Define an equation by self = s.

INPUT:

  • s – a lazy series

EXAMPLES:

We begin by constructing the Catalan numbers:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: C = L.undefined()
sage: C.define(1 + z*C^2)
sage: C
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)
sage: binomial(2000, 1000) / C[1000]                                        # needs sage.symbolic
1001
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> C = L.undefined()
>>> C.define(Integer(1) + z*C**Integer(2))
>>> C
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)
>>> binomial(Integer(2000), Integer(1000)) / C[Integer(1000)]                                        # needs sage.symbolic
1001

The Catalan numbers but with a valuation \(1\):

sage: B = L.undefined(valuation=1)
sage: B.define(z + B^2)
sage: B
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)
>>> from sage.all import *
>>> B = L.undefined(valuation=Integer(1))
>>> B.define(z + B**Integer(2))
>>> B
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

We can define multiple series that are linked:

sage: s = L.undefined()
sage: t = L.undefined()
sage: s.define(1 + z*t^3)
sage: t.define(1 + z*s^2)
sage: s[0:9]
[1, 1, 3, 9, 34, 132, 546, 2327, 10191]
sage: t[0:9]
[1, 1, 2, 7, 24, 95, 386, 1641, 7150]
>>> from sage.all import *
>>> s = L.undefined()
>>> t = L.undefined()
>>> s.define(Integer(1) + z*t**Integer(3))
>>> t.define(Integer(1) + z*s**Integer(2))
>>> s[Integer(0):Integer(9)]
[1, 1, 3, 9, 34, 132, 546, 2327, 10191]
>>> t[Integer(0):Integer(9)]
[1, 1, 2, 7, 24, 95, 386, 1641, 7150]

A bigger example:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: A = L.undefined(valuation=5)
sage: B = L.undefined()
sage: C = L.undefined(valuation=2)
sage: A.define(z^5 + B^2)
sage: B.define(z^5 + C^2)
sage: C.define(z^2 + C^2 + A^2)
sage: A[0:15]
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 5, 4, 14, 10, 48]
sage: B[0:15]
[0, 0, 0, 0, 1, 1, 2, 0, 5, 0, 14, 0, 44, 0, 138]
sage: C[0:15]
[0, 0, 1, 0, 1, 0, 2, 0, 5, 0, 15, 0, 44, 2, 142]
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> A = L.undefined(valuation=Integer(5))
>>> B = L.undefined()
>>> C = L.undefined(valuation=Integer(2))
>>> A.define(z**Integer(5) + B**Integer(2))
>>> B.define(z**Integer(5) + C**Integer(2))
>>> C.define(z**Integer(2) + C**Integer(2) + A**Integer(2))
>>> A[Integer(0):Integer(15)]
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 5, 4, 14, 10, 48]
>>> B[Integer(0):Integer(15)]
[0, 0, 0, 0, 1, 1, 2, 0, 5, 0, 14, 0, 44, 0, 138]
>>> C[Integer(0):Integer(15)]
[0, 0, 1, 0, 1, 0, 2, 0, 5, 0, 15, 0, 44, 2, 142]

Counting binary trees:

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: s = L.undefined(valuation=1)
sage: s.define(z + (s^2+s(z^2))/2)
sage: s[0:9]
[0, 1, 1, 1, 2, 3, 6, 11, 23]
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> s = L.undefined(valuation=Integer(1))
>>> s.define(z + (s**Integer(2)+s(z**Integer(2)))/Integer(2))
>>> s[Integer(0):Integer(9)]
[0, 1, 1, 1, 2, 3, 6, 11, 23]

The \(q\)-Catalan numbers:

sage: R.<q> = ZZ[]
sage: L.<z> = LazyLaurentSeriesRing(R)
sage: s = L.undefined(valuation=0)
sage: s.define(1+z*s*s(q*z))
sage: s
1 + z + (q + 1)*z^2 + (q^3 + q^2 + 2*q + 1)*z^3
 + (q^6 + q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 3*q + 1)*z^4
 + (q^10 + q^9 + 2*q^8 + 3*q^7 + 5*q^6 + 5*q^5 + 7*q^4 + 7*q^3 + 6*q^2 + 4*q + 1)*z^5
 + (q^15 + q^14 + 2*q^13 + 3*q^12 + 5*q^11 + 7*q^10 + 9*q^9 + 11*q^8
    + 14*q^7 + 16*q^6 + 16*q^5 + 17*q^4 + 14*q^3 + 10*q^2 + 5*q + 1)*z^6 + O(z^7)
>>> from sage.all import *
>>> R = ZZ['q']; (q,) = R._first_ngens(1)
>>> L = LazyLaurentSeriesRing(R, names=('z',)); (z,) = L._first_ngens(1)
>>> s = L.undefined(valuation=Integer(0))
>>> s.define(Integer(1)+z*s*s(q*z))
>>> s
1 + z + (q + 1)*z^2 + (q^3 + q^2 + 2*q + 1)*z^3
 + (q^6 + q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 3*q + 1)*z^4
 + (q^10 + q^9 + 2*q^8 + 3*q^7 + 5*q^6 + 5*q^5 + 7*q^4 + 7*q^3 + 6*q^2 + 4*q + 1)*z^5
 + (q^15 + q^14 + 2*q^13 + 3*q^12 + 5*q^11 + 7*q^10 + 9*q^9 + 11*q^8
    + 14*q^7 + 16*q^6 + 16*q^5 + 17*q^4 + 14*q^3 + 10*q^2 + 5*q + 1)*z^6 + O(z^7)

We count unlabeled ordered trees by total number of nodes and number of internal nodes:

sage: R.<q> = QQ[]
sage: Q.<z> = LazyPowerSeriesRing(R)
sage: leaf = z
sage: internal_node = q * z
sage: L = Q(constant=1, degree=1)
sage: T = Q.undefined(valuation=1)
sage: T.define(leaf + internal_node * L(T))
sage: T[0:6]
[0, 1, q, q^2 + q, q^3 + 3*q^2 + q, q^4 + 6*q^3 + 6*q^2 + q]
>>> from sage.all import *
>>> R = QQ['q']; (q,) = R._first_ngens(1)
>>> Q = LazyPowerSeriesRing(R, names=('z',)); (z,) = Q._first_ngens(1)
>>> leaf = z
>>> internal_node = q * z
>>> L = Q(constant=Integer(1), degree=Integer(1))
>>> T = Q.undefined(valuation=Integer(1))
>>> T.define(leaf + internal_node * L(T))
>>> T[Integer(0):Integer(6)]
[0, 1, q, q^2 + q, q^3 + 3*q^2 + q, q^4 + 6*q^3 + 6*q^2 + q]

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: g = L(constant=1, valuation=2)
sage: F = L.undefined()
sage: F.define(1 + g*F)
sage: F[:16]
[1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3]
sage: oeis(_)                                                       # optional - internet
0: A002033: Number of perfect partitions of n.
1: A074206: Kalmár's [Kalmar's] problem: number of ordered factorizations of n.
...

sage: F = L.undefined()
sage: F.define(1 + g*F*F)
sage: F[:16]
[1, 1, 1, 3, 1, 5, 1, 10, 3, 5, 1, 24, 1, 5, 5]
>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, "z")
>>> g = L(constant=Integer(1), valuation=Integer(2))
>>> F = L.undefined()
>>> F.define(Integer(1) + g*F)
>>> F[:Integer(16)]
[1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3]
>>> oeis(_)                                                       # optional - internet
0: A002033: Number of perfect partitions of n.
1: A074206: Kalmár's [Kalmar's] problem: number of ordered factorizations of n.
...

>>> F = L.undefined()
>>> F.define(Integer(1) + g*F*F)
>>> F[:Integer(16)]
[1, 1, 1, 3, 1, 5, 1, 10, 3, 5, 1, 24, 1, 5, 5]

We can compute the Frobenius character of unlabeled trees:

sage: # needs sage.combinat
sage: m = SymmetricFunctions(QQ).m()
sage: s = SymmetricFunctions(QQ).s()
sage: L = LazySymmetricFunctions(m)
sage: E = L(lambda n: s[n], valuation=0)
sage: X = L(s[1])
sage: A = L.undefined()
sage: A.define(X*E(A))
sage: A[:6]
[m[1],
 2*m[1, 1] + m[2],
 9*m[1, 1, 1] + 5*m[2, 1] + 2*m[3],
 64*m[1, 1, 1, 1] + 34*m[2, 1, 1] + 18*m[2, 2] + 13*m[3, 1] + 4*m[4],
 625*m[1, 1, 1, 1, 1] + 326*m[2, 1, 1, 1] + 171*m[2, 2, 1] + 119*m[3, 1, 1] + 63*m[3, 2] + 35*m[4, 1] + 9*m[5]]
>>> from sage.all import *
>>> # needs sage.combinat
>>> m = SymmetricFunctions(QQ).m()
>>> s = SymmetricFunctions(QQ).s()
>>> L = LazySymmetricFunctions(m)
>>> E = L(lambda n: s[n], valuation=Integer(0))
>>> X = L(s[Integer(1)])
>>> A = L.undefined()
>>> A.define(X*E(A))
>>> A[:Integer(6)]
[m[1],
 2*m[1, 1] + m[2],
 9*m[1, 1, 1] + 5*m[2, 1] + 2*m[3],
 64*m[1, 1, 1, 1] + 34*m[2, 1, 1] + 18*m[2, 2] + 13*m[3, 1] + 4*m[4],
 625*m[1, 1, 1, 1, 1] + 326*m[2, 1, 1, 1] + 171*m[2, 2, 1] + 119*m[3, 1, 1] + 63*m[3, 2] + 35*m[4, 1] + 9*m[5]]
shift(n)[source]

Return self with the indices shifted by n.

For example, a Laurent series is multiplied by the power \(z^n\), where \(z\) is the variable of self. For series with a fixed minimal valuation (e.g., power series), this removes any terms that are less than the minimal valuation.

INPUT:

  • n – the amount to shift

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1 / (1 + 2*z)
sage: f
1 - 2*z + 4*z^2 - 8*z^3 + 16*z^4 - 32*z^5 + 64*z^6 + O(z^7)
sage: f.shift(3)
z^3 - 2*z^4 + 4*z^5 - 8*z^6 + 16*z^7 - 32*z^8 + 64*z^9 + O(z^10)
sage: f << -3  # shorthand
z^-3 - 2*z^-2 + 4*z^-1 - 8 + 16*z - 32*z^2 + 64*z^3 + O(z^4)
sage: g = z^-3 + 3 + z^2
sage: g.shift(5)
z^2 + 3*z^5 + z^7
sage: L([2,0,3], valuation=2, degree=7, constant=1) << -2
2 + 3*z^2 + z^5 + z^6 + z^7 + O(z^8)

sage: D = LazyDirichletSeriesRing(QQ, 't')
sage: f = D([0,1,2])
sage: f                                                                     # needs sage.symbolic
1/(2^t) + 2/3^t
sage: sf = f.shift(3)
sage: sf                                                                    # needs sage.symbolic
1/(5^t) + 2/6^t
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = Integer(1) / (Integer(1) + Integer(2)*z)
>>> f
1 - 2*z + 4*z^2 - 8*z^3 + 16*z^4 - 32*z^5 + 64*z^6 + O(z^7)
>>> f.shift(Integer(3))
z^3 - 2*z^4 + 4*z^5 - 8*z^6 + 16*z^7 - 32*z^8 + 64*z^9 + O(z^10)
>>> f << -Integer(3)  # shorthand
z^-3 - 2*z^-2 + 4*z^-1 - 8 + 16*z - 32*z^2 + 64*z^3 + O(z^4)
>>> g = z**-Integer(3) + Integer(3) + z**Integer(2)
>>> g.shift(Integer(5))
z^2 + 3*z^5 + z^7
>>> L([Integer(2),Integer(0),Integer(3)], valuation=Integer(2), degree=Integer(7), constant=Integer(1)) << -Integer(2)
2 + 3*z^2 + z^5 + z^6 + z^7 + O(z^8)

>>> D = LazyDirichletSeriesRing(QQ, 't')
>>> f = D([Integer(0),Integer(1),Integer(2)])
>>> f                                                                     # needs sage.symbolic
1/(2^t) + 2/3^t
>>> sf = f.shift(Integer(3))
>>> sf                                                                    # needs sage.symbolic
1/(5^t) + 2/6^t

Examples with power series (where the minimal valuation is \(0\)):

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: f = 1 / (1 - x)
sage: f.shift(2)
x^2 + x^3 + x^4 + O(x^5)
sage: g = f.shift(-1); g
1 + x + x^2 + O(x^3)
sage: f == g
True
sage: g[-1]
0
sage: h = L(lambda n: 1)
sage: LazyPowerSeriesRing.options.halting_precision(20)  # verify up to degree 20
sage: f == h
True
sage: h == f
True
sage: h.shift(-1) == h
True
sage: LazyPowerSeriesRing.options._reset()

sage: fp = L([3,3,3], constant=1)
sage: fp.shift(2)
3*x^2 + 3*x^3 + 3*x^4 + x^5 + x^6 + x^7 + O(x^8)
sage: fp.shift(-2)
3 + x + x^2 + x^3 + O(x^4)
sage: fp.shift(-7)
1 + x + x^2 + O(x^3)
sage: fp.shift(-5) == g
True
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x',)); (x,) = L._first_ngens(1)
>>> f = Integer(1) / (Integer(1) - x)
>>> f.shift(Integer(2))
x^2 + x^3 + x^4 + O(x^5)
>>> g = f.shift(-Integer(1)); g
1 + x + x^2 + O(x^3)
>>> f == g
True
>>> g[-Integer(1)]
0
>>> h = L(lambda n: Integer(1))
>>> LazyPowerSeriesRing.options.halting_precision(Integer(20))  # verify up to degree 20
>>> f == h
True
>>> h == f
True
>>> h.shift(-Integer(1)) == h
True
>>> LazyPowerSeriesRing.options._reset()

>>> fp = L([Integer(3),Integer(3),Integer(3)], constant=Integer(1))
>>> fp.shift(Integer(2))
3*x^2 + 3*x^3 + 3*x^4 + x^5 + x^6 + x^7 + O(x^8)
>>> fp.shift(-Integer(2))
3 + x + x^2 + x^3 + O(x^4)
>>> fp.shift(-Integer(7))
1 + x + x^2 + O(x^3)
>>> fp.shift(-Integer(5)) == g
True

We compare the shifting with converting to the fraction field (see also Issue #35293):

sage: M = L.fraction_field()
sage: f = L([1,2,3,4]); f
1 + 2*x + 3*x^2 + 4*x^3
sage: f.shift(-3)
4
sage: M(f).shift(-3)
x^-3 + 2*x^-2 + 3*x^-1 + 4
>>> from sage.all import *
>>> M = L.fraction_field()
>>> f = L([Integer(1),Integer(2),Integer(3),Integer(4)]); f
1 + 2*x + 3*x^2 + 4*x^3
>>> f.shift(-Integer(3))
4
>>> M(f).shift(-Integer(3))
x^-3 + 2*x^-2 + 3*x^-1 + 4

An example with a more general function:

sage: fun = lambda n: 1 if ZZ(n).is_power_of(2) else 0
sage: f = L(fun); f
x + x^2 + x^4 + O(x^7)
sage: fs = f.shift(-4)
sage: fs
1 + x^4 + O(x^7)
sage: fs.shift(4)
x^4 + x^8 + O(x^11)
sage: M(f).shift(-4)
x^-3 + x^-2 + 1 + O(x^4)
>>> from sage.all import *
>>> fun = lambda n: Integer(1) if ZZ(n).is_power_of(Integer(2)) else Integer(0)
>>> f = L(fun); f
x + x^2 + x^4 + O(x^7)
>>> fs = f.shift(-Integer(4))
>>> fs
1 + x^4 + O(x^7)
>>> fs.shift(Integer(4))
x^4 + x^8 + O(x^11)
>>> M(f).shift(-Integer(4))
x^-3 + x^-2 + 1 + O(x^4)
sin()[source]

Return the sine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sin(z)
z - 1/6*z^3 + 1/120*z^5 - 1/5040*z^7 + O(z^8)

sage: sin(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sin(x/(1-y)).polynomial(3)
-1/6*x^3 + x*y^2 + x*y + x
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> sin(z)
z - 1/6*z^3 + 1/120*z^5 - 1/5040*z^7 + O(z^8)

>>> sin(Integer(1) + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> sin(x/(Integer(1)-y)).polynomial(Integer(3))
-1/6*x^3 + x*y^2 + x*y + x
sinh()[source]

Return the hyperbolic sine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sinh(z)
z + 1/6*z^3 + 1/120*z^5 + 1/5040*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sinh(x/(1-y))
x + x*y + (1/6*x^3+x*y^2) + (1/2*x^3*y+x*y^3)
 + (1/120*x^5+x^3*y^2+x*y^4) + (1/24*x^5*y+5/3*x^3*y^3+x*y^5)
 + (1/5040*x^7+1/8*x^5*y^2+5/2*x^3*y^4+x*y^6) + O(x,y)^8
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> sinh(z)
z + 1/6*z^3 + 1/120*z^5 + 1/5040*z^7 + O(z^8)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> sinh(x/(Integer(1)-y))
x + x*y + (1/6*x^3+x*y^2) + (1/2*x^3*y+x*y^3)
 + (1/120*x^5+x^3*y^2+x*y^4) + (1/24*x^5*y+5/3*x^3*y^3+x*y^5)
 + (1/5040*x^7+1/8*x^5*y^2+5/2*x^3*y^4+x*y^6) + O(x,y)^8
sqrt()[source]

Return self^(1/2).

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sqrt(1+z)
1 + 1/2*z - 1/8*z^2 + 1/16*z^3 - 5/128*z^4 + 7/256*z^5 - 21/1024*z^6 + O(z^7)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: sqrt(1+x/(1-y))
1 + 1/2*x + (-1/8*x^2+1/2*x*y) + (1/16*x^3-1/4*x^2*y+1/2*x*y^2)
 + (-5/128*x^4+3/16*x^3*y-3/8*x^2*y^2+1/2*x*y^3)
 + (7/256*x^5-5/32*x^4*y+3/8*x^3*y^2-1/2*x^2*y^3+1/2*x*y^4)
 + (-21/1024*x^6+35/256*x^5*y-25/64*x^4*y^2+5/8*x^3*y^3-5/8*x^2*y^4+1/2*x*y^5)
 + O(x,y)^7
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> sqrt(Integer(1)+z)
1 + 1/2*z - 1/8*z^2 + 1/16*z^3 - 5/128*z^4 + 7/256*z^5 - 21/1024*z^6 + O(z^7)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> sqrt(Integer(1)+x/(Integer(1)-y))
1 + 1/2*x + (-1/8*x^2+1/2*x*y) + (1/16*x^3-1/4*x^2*y+1/2*x*y^2)
 + (-5/128*x^4+3/16*x^3*y-3/8*x^2*y^2+1/2*x*y^3)
 + (7/256*x^5-5/32*x^4*y+3/8*x^3*y^2-1/2*x^2*y^3+1/2*x*y^4)
 + (-21/1024*x^6+35/256*x^5*y-25/64*x^4*y^2+5/8*x^3*y^3-5/8*x^2*y^4+1/2*x*y^5)
 + O(x,y)^7

This also works for Dirichlet series:

sage: # needs sage.symbolic
sage: D = LazyDirichletSeriesRing(SR, "s")
sage: Z = D(constant=1)
sage: f = sqrt(Z);  f
1 + 1/2/2^s + 1/2/3^s + 3/8/4^s + 1/2/5^s + 1/4/6^s + 1/2/7^s + O(1/(8^s))
sage: f*f - Z
O(1/(8^s))
>>> from sage.all import *
>>> # needs sage.symbolic
>>> D = LazyDirichletSeriesRing(SR, "s")
>>> Z = D(constant=Integer(1))
>>> f = sqrt(Z);  f
1 + 1/2/2^s + 1/2/3^s + 3/8/4^s + 1/2/5^s + 1/4/6^s + 1/2/7^s + O(1/(8^s))
>>> f*f - Z
O(1/(8^s))
tan()[source]

Return the tangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: tan(z)
z + 1/3*z^3 + 2/15*z^5 + 17/315*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: tan(x/(1-y)).polynomial(5)
2/15*x^5 + 2*x^3*y^2 + x*y^4 + x^3*y + x*y^3 + 1/3*x^3 + x*y^2 + x*y + x
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> tan(z)
z + 1/3*z^3 + 2/15*z^5 + 17/315*z^7 + O(z^8)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> tan(x/(Integer(1)-y)).polynomial(Integer(5))
2/15*x^5 + 2*x^3*y^2 + x*y^4 + x^3*y + x*y^3 + 1/3*x^3 + x*y^2 + x*y + x
tanh()[source]

Return the hyperbolic tangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: tanh(z)                                                               # needs sage.libs.flint
z - 1/3*z^3 + 2/15*z^5 - 17/315*z^7 + O(z^8)

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: tanh(x/(1-y))                                                         # needs sage.libs.flint
x + x*y + (-1/3*x^3+x*y^2) + (-x^3*y+x*y^3) + (2/15*x^5-2*x^3*y^2+x*y^4)
 + (2/3*x^5*y-10/3*x^3*y^3+x*y^5) + (-17/315*x^7+2*x^5*y^2-5*x^3*y^4+x*y^6) + O(x,y)^8
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> tanh(z)                                                               # needs sage.libs.flint
z - 1/3*z^3 + 2/15*z^5 - 17/315*z^7 + O(z^8)

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> tanh(x/(Integer(1)-y))                                                         # needs sage.libs.flint
x + x*y + (-1/3*x^3+x*y^2) + (-x^3*y+x*y^3) + (2/15*x^5-2*x^3*y^2+x*y^4)
 + (2/3*x^5*y-10/3*x^3*y^3+x*y^5) + (-17/315*x^7+2*x^5*y^2-5*x^3*y^4+x*y^6) + O(x,y)^8
truncate(d)[source]

Return the series obtained by removing all terms of degree at least d.

INPUT:

  • d – integer; the degree from which the series is truncated

EXAMPLES:

Dense implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: alpha = 1/(1-z)
sage: alpha
1 + z + z^2 + O(z^3)
sage: beta = alpha.truncate(5)
sage: beta
1 + z + z^2 + z^3 + z^4
sage: alpha - beta
z^5 + z^6 + z^7 + O(z^8)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.truncate(4)
z + 2*z^2 + 3*z^3
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, sparse=False, names=('z',)); (z,) = L._first_ngens(1)
>>> alpha = Integer(1)/(Integer(1)-z)
>>> alpha
1 + z + z^2 + O(z^3)
>>> beta = alpha.truncate(Integer(5))
>>> beta
1 + z + z^2 + z^3 + z^4
>>> alpha - beta
z^5 + z^6 + z^7 + O(z^8)
>>> M = L(lambda n: n, valuation=Integer(0)); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
>>> M.truncate(Integer(4))
z + 2*z^2 + 3*z^3

Sparse Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.truncate(4)
z + 2*z^2 + 3*z^3
>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, sparse=True, names=('z',)); (z,) = L._first_ngens(1)
>>> M = L(lambda n: n, valuation=Integer(0)); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
>>> M.truncate(Integer(4))
z + 2*z^2 + 3*z^3

Series which are known to be exact can also be truncated:

sage: M = z + z^2 + z^3 + z^4
sage: M.truncate(4)
z + z^2 + z^3
>>> from sage.all import *
>>> M = z + z**Integer(2) + z**Integer(3) + z**Integer(4)
>>> M.truncate(Integer(4))
z + z^2 + z^3
class sage.rings.lazy_series.LazyPowerSeries(parent, coeff_stream)[source]

Bases: LazyCauchyProductSeries

A Taylor series where the coefficients are computed lazily.

EXAMPLES:

sage: L.<x, y> = LazyPowerSeriesRing(ZZ)
sage: f = 1 / (1 - x^2 + y^3); f
1 + x^2 + (-y^3) + x^4 + (-2*x^2*y^3) + (x^6+y^6) + O(x,y)^7
sage: P.<x, y> = PowerSeriesRing(ZZ, default_prec=101)
sage: g = 1 / (1 - x^2 + y^3); f[100] - g[100]
0
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> f = Integer(1) / (Integer(1) - x**Integer(2) + y**Integer(3)); f
1 + x^2 + (-y^3) + x^4 + (-2*x^2*y^3) + (x^6+y^6) + O(x,y)^7
>>> P = PowerSeriesRing(ZZ, default_prec=Integer(101), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> g = Integer(1) / (Integer(1) - x**Integer(2) + y**Integer(3)); f[Integer(100)] - g[Integer(100)]
0

Lazy Taylor series is picklable:

sage: g = loads(dumps(f))
sage: g
1 + x^2 + (-y^3) + x^4 + (-2*x^2*y^3) + (x^6+y^6) + O(x,y)^7
sage: g == f
True
>>> from sage.all import *
>>> g = loads(dumps(f))
>>> g
1 + x^2 + (-y^3) + x^4 + (-2*x^2*y^3) + (x^6+y^6) + O(x,y)^7
>>> g == f
True
O(prec)[source]

Return the power series of precision at most prec obtained by adding \(O(q^\text{prec})\) to \(f\), where \(q\) is the (tuple of) variable(s).

EXAMPLES:

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: f = 1 / (1 - x + y)
sage: f
1 + (x-y) + (x^2-2*x*y+y^2) + (x^3-3*x^2*y+3*x*y^2-y^3)
 + (x^4-4*x^3*y+6*x^2*y^2-4*x*y^3+y^4)
 + (x^5-5*x^4*y+10*x^3*y^2-10*x^2*y^3+5*x*y^4-y^5)
 + (x^6-6*x^5*y+15*x^4*y^2-20*x^3*y^3+15*x^2*y^4-6*x*y^5+y^6)
 + O(x,y)^7
sage: f3 = f.add_bigoh(3); f3
1 + x - y + x^2 - 2*x*y + y^2 + O(x, y)^3
sage: f3.parent()
Multivariate Power Series Ring in x, y over Rational Field

sage: R.<t> = QQ[]
sage: L.<x> = LazyPowerSeriesRing(R)
sage: f = 1 / (1 - t^3*x)
sage: f
1 + t^3*x + t^6*x^2 + t^9*x^3 + t^12*x^4 + t^15*x^5 + t^18*x^6 + O(x^7)
sage: f3 = f.add_bigoh(3); f3
1 + t^3*x + t^6*x^2 + O(x^3)
sage: f3.parent()
Power Series Ring in x over Univariate Polynomial Ring in t
 over Rational Field
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> f = Integer(1) / (Integer(1) - x + y)
>>> f
1 + (x-y) + (x^2-2*x*y+y^2) + (x^3-3*x^2*y+3*x*y^2-y^3)
 + (x^4-4*x^3*y+6*x^2*y^2-4*x*y^3+y^4)
 + (x^5-5*x^4*y+10*x^3*y^2-10*x^2*y^3+5*x*y^4-y^5)
 + (x^6-6*x^5*y+15*x^4*y^2-20*x^3*y^3+15*x^2*y^4-6*x*y^5+y^6)
 + O(x,y)^7
>>> f3 = f.add_bigoh(Integer(3)); f3
1 + x - y + x^2 - 2*x*y + y^2 + O(x, y)^3
>>> f3.parent()
Multivariate Power Series Ring in x, y over Rational Field

>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> L = LazyPowerSeriesRing(R, names=('x',)); (x,) = L._first_ngens(1)
>>> f = Integer(1) / (Integer(1) - t**Integer(3)*x)
>>> f
1 + t^3*x + t^6*x^2 + t^9*x^3 + t^12*x^4 + t^15*x^5 + t^18*x^6 + O(x^7)
>>> f3 = f.add_bigoh(Integer(3)); f3
1 + t^3*x + t^6*x^2 + O(x^3)
>>> f3.parent()
Power Series Ring in x over Univariate Polynomial Ring in t
 over Rational Field
adams_operator(p)[source]

Return the image of self under the Adams operator of index p.

This raises all variables to the power p, both the power series variables and the variables inside the coefficient ring.

INPUT:

  • p – positive integer

EXAMPLES:

With no variables in the base ring:

sage: A = LazyPowerSeriesRing(QQ,'t')
sage: f = A([1,2,3,4]); f
1 + 2*t + 3*t^2 + 4*t^3
sage: f.adams_operator(2)
1 + 2*t^2 + 3*t^4 + 4*t^6
>>> from sage.all import *
>>> A = LazyPowerSeriesRing(QQ,'t')
>>> f = A([Integer(1),Integer(2),Integer(3),Integer(4)]); f
1 + 2*t + 3*t^2 + 4*t^3
>>> f.adams_operator(Integer(2))
1 + 2*t^2 + 3*t^4 + 4*t^6

With variables in the base ring:

sage: q = polygen(QQ,'q')
sage: A = LazyPowerSeriesRing(q.parent(),'t')
sage: f = A([0,1+q,2,3+q**2]); f
((q+1)*t) + 2*t^2 + ((q^2+3)*t^3)
sage: f.adams_operator(2)
((q^2+1)*t^2) + 2*t^4 + ((q^4+3)*t^6)
>>> from sage.all import *
>>> q = polygen(QQ,'q')
>>> A = LazyPowerSeriesRing(q.parent(),'t')
>>> f = A([Integer(0),Integer(1)+q,Integer(2),Integer(3)+q**Integer(2)]); f
((q+1)*t) + 2*t^2 + ((q^2+3)*t^3)
>>> f.adams_operator(Integer(2))
((q^2+1)*t^2) + 2*t^4 + ((q^4+3)*t^6)

In the multivariate case:

sage: A = LazyPowerSeriesRing(ZZ,'t,u')
sage: f = A({(1,2):4,(2,3):6}); f
4*t*u^2 + 6*t^2*u^3
sage: f.adams_operator(3)
4*t^3*u^6 + 6*t^6*u^9
>>> from sage.all import *
>>> A = LazyPowerSeriesRing(ZZ,'t,u')
>>> f = A({(Integer(1),Integer(2)):Integer(4),(Integer(2),Integer(3)):Integer(6)}); f
4*t*u^2 + 6*t^2*u^3
>>> f.adams_operator(Integer(3))
4*t^3*u^6 + 6*t^6*u^9
add_bigoh(prec)[source]

Return the power series of precision at most prec obtained by adding \(O(q^\text{prec})\) to \(f\), where \(q\) is the (tuple of) variable(s).

EXAMPLES:

sage: L.<x,y> = LazyPowerSeriesRing(QQ)
sage: f = 1 / (1 - x + y)
sage: f
1 + (x-y) + (x^2-2*x*y+y^2) + (x^3-3*x^2*y+3*x*y^2-y^3)
 + (x^4-4*x^3*y+6*x^2*y^2-4*x*y^3+y^4)
 + (x^5-5*x^4*y+10*x^3*y^2-10*x^2*y^3+5*x*y^4-y^5)
 + (x^6-6*x^5*y+15*x^4*y^2-20*x^3*y^3+15*x^2*y^4-6*x*y^5+y^6)
 + O(x,y)^7
sage: f3 = f.add_bigoh(3); f3
1 + x - y + x^2 - 2*x*y + y^2 + O(x, y)^3
sage: f3.parent()
Multivariate Power Series Ring in x, y over Rational Field

sage: R.<t> = QQ[]
sage: L.<x> = LazyPowerSeriesRing(R)
sage: f = 1 / (1 - t^3*x)
sage: f
1 + t^3*x + t^6*x^2 + t^9*x^3 + t^12*x^4 + t^15*x^5 + t^18*x^6 + O(x^7)
sage: f3 = f.add_bigoh(3); f3
1 + t^3*x + t^6*x^2 + O(x^3)
sage: f3.parent()
Power Series Ring in x over Univariate Polynomial Ring in t
 over Rational Field
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> f = Integer(1) / (Integer(1) - x + y)
>>> f
1 + (x-y) + (x^2-2*x*y+y^2) + (x^3-3*x^2*y+3*x*y^2-y^3)
 + (x^4-4*x^3*y+6*x^2*y^2-4*x*y^3+y^4)
 + (x^5-5*x^4*y+10*x^3*y^2-10*x^2*y^3+5*x*y^4-y^5)
 + (x^6-6*x^5*y+15*x^4*y^2-20*x^3*y^3+15*x^2*y^4-6*x*y^5+y^6)
 + O(x,y)^7
>>> f3 = f.add_bigoh(Integer(3)); f3
1 + x - y + x^2 - 2*x*y + y^2 + O(x, y)^3
>>> f3.parent()
Multivariate Power Series Ring in x, y over Rational Field

>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> L = LazyPowerSeriesRing(R, names=('x',)); (x,) = L._first_ngens(1)
>>> f = Integer(1) / (Integer(1) - t**Integer(3)*x)
>>> f
1 + t^3*x + t^6*x^2 + t^9*x^3 + t^12*x^4 + t^15*x^5 + t^18*x^6 + O(x^7)
>>> f3 = f.add_bigoh(Integer(3)); f3
1 + t^3*x + t^6*x^2 + O(x^3)
>>> f3.parent()
Power Series Ring in x over Univariate Polynomial Ring in t
 over Rational Field
compose(*g)[source]

Return the composition of self with g.

The arity of self must be equal to the number of arguments provided.

Given a Taylor series \(f\) of arity \(n\) and a tuple of Taylor series \(g = (g_1,\dots, g_n)\) over the same base ring, the composition \(f \circ g\) is defined if and only if for each \(1\leq i\leq n\):

  • \(g_i\) is zero, or

  • setting all variables except the \(i\)-th in \(f\) to zero yields a polynomial, or

  • \(\mathrm{val}(g_i) > 0\).

If \(f\) is a univariate ‘exact’ series, we can check whether \(f\) is a actually a polynomial. However, if \(f\) is a multivariate series, we have no way to test whether setting all but one variable of \(f\) to zero yields a polynomial, except if \(f\) itself is ‘exact’ and therefore a polynomial.

INPUT:

  • g – other series, all can be coerced into the same parent

EXAMPLES:

sage: L.<x, y, z> = LazyPowerSeriesRing(QQ)
sage: M.<a, b> = LazyPowerSeriesRing(ZZ)
sage: g1 = 1 / (1 - x)
sage: g2 = x + y^2
sage: p = a^2 + b + 1
sage: p(g1, g2) - g1^2 - g2 - 1
O(x,y,z)^7
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> M = LazyPowerSeriesRing(ZZ, names=('a', 'b',)); (a, b,) = M._first_ngens(2)
>>> g1 = Integer(1) / (Integer(1) - x)
>>> g2 = x + y**Integer(2)
>>> p = a**Integer(2) + b + Integer(1)
>>> p(g1, g2) - g1**Integer(2) - g2 - Integer(1)
O(x,y,z)^7

The number of mappings from a set with \(m\) elements to a set with \(n\) elements:

sage: M.<a> = LazyPowerSeriesRing(QQ)
sage: Ea = M(lambda n: 1/factorial(n))
sage: Ex = L(lambda n: 1/factorial(n)*x^n)
sage: Ea(Ex*y)[5]
1/24*x^4*y + 2/3*x^3*y^2 + 3/4*x^2*y^3 + 1/6*x*y^4 + 1/120*y^5
>>> from sage.all import *
>>> M = LazyPowerSeriesRing(QQ, names=('a',)); (a,) = M._first_ngens(1)
>>> Ea = M(lambda n: Integer(1)/factorial(n))
>>> Ex = L(lambda n: Integer(1)/factorial(n)*x**n)
>>> Ea(Ex*y)[Integer(5)]
1/24*x^4*y + 2/3*x^3*y^2 + 3/4*x^2*y^3 + 1/6*x*y^4 + 1/120*y^5

So, there are \(3! 2! 2/3 = 8\) mappings from a three element set to a two element set.

We perform the composition with a lazy Laurent series:

sage: N.<w> = LazyLaurentSeriesRing(QQ)
sage: f1 = 1 / (1 - w)
sage: f2 = cot(w / (1 - w))
sage: p(f1, f2)
w^-1 + 1 + 5/3*w + 8/3*w^2 + 164/45*w^3 + 23/5*w^4 + 5227/945*w^5 + O(w^6)
>>> from sage.all import *
>>> N = LazyLaurentSeriesRing(QQ, names=('w',)); (w,) = N._first_ngens(1)
>>> f1 = Integer(1) / (Integer(1) - w)
>>> f2 = cot(w / (Integer(1) - w))
>>> p(f1, f2)
w^-1 + 1 + 5/3*w + 8/3*w^2 + 164/45*w^3 + 23/5*w^4 + 5227/945*w^5 + O(w^6)

We perform the composition with a lazy Dirichlet series:

sage: # needs sage.symbolic
sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: g = D(constant=1)-1
sage: g
1/(2^s) + 1/(3^s) + 1/(4^s) + O(1/(5^s))
sage: f = 1 / (1 - x - y*z); f
1 + x + (x^2+y*z) + (x^3+2*x*y*z) + (x^4+3*x^2*y*z+y^2*z^2)
 + (x^5+4*x^3*y*z+3*x*y^2*z^2)
 + (x^6+5*x^4*y*z+6*x^2*y^2*z^2+y^3*z^3)
 + O(x,y,z)^7
sage: fog = f(g, g, g)
sage: fog
1 + 1/(2^s) + 1/(3^s) + 3/4^s + 1/(5^s) + 5/6^s + O(1/(7^s))
sage: fg = 1 / (1 - g - g*g)
sage: fg
1 + 1/(2^s) + 1/(3^s) + 3/4^s + 1/(5^s) + 5/6^s + 1/(7^s) + O(1/(8^s))
sage: fog - fg
O(1/(8^s))

sage: f = 1 / (1 - 2*a)
sage: f(g)                                                                  # needs sage.symbolic
1 + 2/2^s + 2/3^s + 6/4^s + 2/5^s + 10/6^s + 2/7^s + O(1/(8^s))
sage: 1 / (1 - 2*g)                                                         # needs sage.symbolic
1 + 2/2^s + 2/3^s + 6/4^s + 2/5^s + 10/6^s + 2/7^s + O(1/(8^s))
>>> from sage.all import *
>>> # needs sage.symbolic
>>> D = LazyDirichletSeriesRing(QQ, "s")
>>> g = D(constant=Integer(1))-Integer(1)
>>> g
1/(2^s) + 1/(3^s) + 1/(4^s) + O(1/(5^s))
>>> f = Integer(1) / (Integer(1) - x - y*z); f
1 + x + (x^2+y*z) + (x^3+2*x*y*z) + (x^4+3*x^2*y*z+y^2*z^2)
 + (x^5+4*x^3*y*z+3*x*y^2*z^2)
 + (x^6+5*x^4*y*z+6*x^2*y^2*z^2+y^3*z^3)
 + O(x,y,z)^7
>>> fog = f(g, g, g)
>>> fog
1 + 1/(2^s) + 1/(3^s) + 3/4^s + 1/(5^s) + 5/6^s + O(1/(7^s))
>>> fg = Integer(1) / (Integer(1) - g - g*g)
>>> fg
1 + 1/(2^s) + 1/(3^s) + 3/4^s + 1/(5^s) + 5/6^s + 1/(7^s) + O(1/(8^s))
>>> fog - fg
O(1/(8^s))

>>> f = Integer(1) / (Integer(1) - Integer(2)*a)
>>> f(g)                                                                  # needs sage.symbolic
1 + 2/2^s + 2/3^s + 6/4^s + 2/5^s + 10/6^s + 2/7^s + O(1/(8^s))
>>> Integer(1) / (Integer(1) - Integer(2)*g)                                                         # needs sage.symbolic
1 + 2/2^s + 2/3^s + 6/4^s + 2/5^s + 10/6^s + 2/7^s + O(1/(8^s))

The output parent is always the common parent between the base ring of \(f\) and the parent of \(g\) or extended to the corresponding lazy series:

sage: T.<x,y> = LazyPowerSeriesRing(QQ)
sage: R.<a,b,c> = ZZ[]
sage: S.<v> = R[]
sage: L.<z> = LaurentPolynomialRing(ZZ)
sage: parent(x(a, b))
Multivariate Polynomial Ring in a, b, c over Rational Field
sage: parent(x(CC(2), a))
Multivariate Polynomial Ring in a, b, c over Complex Field with 53 bits of precision
sage: parent(x(0, 0))
Rational Field
sage: f = 1 / (1 - x - y); f
1 + (x+y) + (x^2+2*x*y+y^2) + (x^3+3*x^2*y+3*x*y^2+y^3)
 + (x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+y^4)
 + (x^5+5*x^4*y+10*x^3*y^2+10*x^2*y^3+5*x*y^4+y^5)
 + (x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5+y^6)
 + O(x,y)^7
sage: f(a^2, b*c)
1 + (a^2+b*c) + (a^4+2*a^2*b*c+b^2*c^2) + (a^6+3*a^4*b*c+3*a^2*b^2*c^2+b^3*c^3) + O(a,b,c)^7
sage: f(v, v^2)
1 + v + 2*v^2 + 3*v^3 + 5*v^4 + 8*v^5 + 13*v^6 + O(v^7)
sage: f(z, z^2 + z)
1 + 2*z + 5*z^2 + 12*z^3 + 29*z^4 + 70*z^5 + 169*z^6 + O(z^7)
sage: three = T(3)(a^2, b); three
3
sage: parent(three)
Multivariate Polynomial Ring in a, b, c over Rational Field
>>> from sage.all import *
>>> T = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = T._first_ngens(2)
>>> R = ZZ['a, b, c']; (a, b, c,) = R._first_ngens(3)
>>> S = R['v']; (v,) = S._first_ngens(1)
>>> L = LaurentPolynomialRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> parent(x(a, b))
Multivariate Polynomial Ring in a, b, c over Rational Field
>>> parent(x(CC(Integer(2)), a))
Multivariate Polynomial Ring in a, b, c over Complex Field with 53 bits of precision
>>> parent(x(Integer(0), Integer(0)))
Rational Field
>>> f = Integer(1) / (Integer(1) - x - y); f
1 + (x+y) + (x^2+2*x*y+y^2) + (x^3+3*x^2*y+3*x*y^2+y^3)
 + (x^4+4*x^3*y+6*x^2*y^2+4*x*y^3+y^4)
 + (x^5+5*x^4*y+10*x^3*y^2+10*x^2*y^3+5*x*y^4+y^5)
 + (x^6+6*x^5*y+15*x^4*y^2+20*x^3*y^3+15*x^2*y^4+6*x*y^5+y^6)
 + O(x,y)^7
>>> f(a**Integer(2), b*c)
1 + (a^2+b*c) + (a^4+2*a^2*b*c+b^2*c^2) + (a^6+3*a^4*b*c+3*a^2*b^2*c^2+b^3*c^3) + O(a,b,c)^7
>>> f(v, v**Integer(2))
1 + v + 2*v^2 + 3*v^3 + 5*v^4 + 8*v^5 + 13*v^6 + O(v^7)
>>> f(z, z**Integer(2) + z)
1 + 2*z + 5*z^2 + 12*z^3 + 29*z^4 + 70*z^5 + 169*z^6 + O(z^7)
>>> three = T(Integer(3))(a**Integer(2), b); three
3
>>> parent(three)
Multivariate Polynomial Ring in a, b, c over Rational Field
compositional_inverse()[source]

Return the compositional inverse of self.

Given a Taylor series \(f\) in one variable, the compositional inverse is a power series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

  • \(\mathrm{val}(f) = 1\), or

  • \(f = a + b z\) with \(a, b \neq 0\).

EXAMPLES:

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: (2*z).revert()
1/2*z
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

sage: s = L(degree=1, constant=-1)
sage: s.revert()
-z - z^2 - z^3 + O(z^4)

sage: s = L(degree=1, constant=1)
sage: s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> (Integer(2)*z).revert()
1/2*z
>>> (z-z**Integer(2)).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

>>> s = L(degree=Integer(1), constant=-Integer(1))
>>> s.revert()
-z - z^2 - z^3 + O(z^4)

>>> s = L(degree=Integer(1), constant=Integer(1))
>>> s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)

Warning

For series not known to be eventually constant (e.g., being defined by a function) with approximate valuation \(\leq 1\) (but not necessarily its true valuation), this assumes that this is the actual valuation:

sage: f = L(lambda n: n if n > 2 else 0)
sage: f.revert()
<repr(... failed: ValueError: generator already executing>
>>> from sage.all import *
>>> f = L(lambda n: n if n > Integer(2) else Integer(0))
>>> f.revert()
<repr(... failed: ValueError: generator already executing>
compute_coefficients(i)[source]

Compute all the coefficients of self up to i.

This method is deprecated, it has no effect anymore.

derivative(*args)[source]

Return the derivative of the Taylor series.

Multiple variables and iteration counts may be supplied; see the documentation of sage.calculus.functional.derivative() function for details.

EXAMPLES:

sage: T.<z> = LazyPowerSeriesRing(ZZ)
sage: z.derivative()
1
sage: (1 + z + z^2).derivative(3)
0
sage: (z^2 + z^4 + z^10).derivative(3)
24*z + 720*z^7
sage: (1 / (1-z)).derivative()
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)
sage: T([1, 1, 1], constant=4).derivative()
1 + 2*z + 12*z^2 + 16*z^3 + 20*z^4 + 24*z^5 + 28*z^6 + O(z^7)

sage: R.<q> = QQ[]
sage: L.<x, y> = LazyPowerSeriesRing(R)
sage: f = 1 / (1-q*x+y); f
1 + (q*x-y) + (q^2*x^2+(-2*q)*x*y+y^2)
 + (q^3*x^3+(-3*q^2)*x^2*y+3*q*x*y^2-y^3)
 + (q^4*x^4+(-4*q^3)*x^3*y+6*q^2*x^2*y^2+(-4*q)*x*y^3+y^4)
 + (q^5*x^5+(-5*q^4)*x^4*y+10*q^3*x^3*y^2+(-10*q^2)*x^2*y^3+5*q*x*y^4-y^5)
 + (q^6*x^6+(-6*q^5)*x^5*y+15*q^4*x^4*y^2+(-20*q^3)*x^3*y^3+15*q^2*x^2*y^4+(-6*q)*x*y^5+y^6)
 + O(x,y)^7
sage: f.derivative(q)
x + (2*q*x^2+(-2)*x*y) + (3*q^2*x^3+(-6*q)*x^2*y+3*x*y^2)
 + (4*q^3*x^4+(-12*q^2)*x^3*y+12*q*x^2*y^2+(-4)*x*y^3)
 + (5*q^4*x^5+(-20*q^3)*x^4*y+30*q^2*x^3*y^2+(-20*q)*x^2*y^3+5*x*y^4)
 + (6*q^5*x^6+(-30*q^4)*x^5*y+60*q^3*x^4*y^2+(-60*q^2)*x^3*y^3+30*q*x^2*y^4+(-6)*x*y^5)
 + O(x,y)^7
>>> from sage.all import *
>>> T = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = T._first_ngens(1)
>>> z.derivative()
1
>>> (Integer(1) + z + z**Integer(2)).derivative(Integer(3))
0
>>> (z**Integer(2) + z**Integer(4) + z**Integer(10)).derivative(Integer(3))
24*z + 720*z^7
>>> (Integer(1) / (Integer(1)-z)).derivative()
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)
>>> T([Integer(1), Integer(1), Integer(1)], constant=Integer(4)).derivative()
1 + 2*z + 12*z^2 + 16*z^3 + 20*z^4 + 24*z^5 + 28*z^6 + O(z^7)

>>> R = QQ['q']; (q,) = R._first_ngens(1)
>>> L = LazyPowerSeriesRing(R, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> f = Integer(1) / (Integer(1)-q*x+y); f
1 + (q*x-y) + (q^2*x^2+(-2*q)*x*y+y^2)
 + (q^3*x^3+(-3*q^2)*x^2*y+3*q*x*y^2-y^3)
 + (q^4*x^4+(-4*q^3)*x^3*y+6*q^2*x^2*y^2+(-4*q)*x*y^3+y^4)
 + (q^5*x^5+(-5*q^4)*x^4*y+10*q^3*x^3*y^2+(-10*q^2)*x^2*y^3+5*q*x*y^4-y^5)
 + (q^6*x^6+(-6*q^5)*x^5*y+15*q^4*x^4*y^2+(-20*q^3)*x^3*y^3+15*q^2*x^2*y^4+(-6*q)*x*y^5+y^6)
 + O(x,y)^7
>>> f.derivative(q)
x + (2*q*x^2+(-2)*x*y) + (3*q^2*x^3+(-6*q)*x^2*y+3*x*y^2)
 + (4*q^3*x^4+(-12*q^2)*x^3*y+12*q*x^2*y^2+(-4)*x*y^3)
 + (5*q^4*x^5+(-20*q^3)*x^4*y+30*q^2*x^3*y^2+(-20*q)*x^2*y^3+5*x*y^4)
 + (6*q^5*x^6+(-30*q^4)*x^5*y+60*q^3*x^4*y^2+(-60*q^2)*x^3*y^3+30*q*x^2*y^4+(-6)*x*y^5)
 + O(x,y)^7

Multivariate:

sage: L.<x,y,z> = LazyPowerSeriesRing(QQ)
sage: f = (x + y^2 + z)^3; f
(x^3+3*x^2*z+3*x*z^2+z^3) + (3*x^2*y^2+6*x*y^2*z+3*y^2*z^2) + (3*x*y^4+3*y^4*z) + y^6
sage: f.derivative(x)
(3*x^2+6*x*z+3*z^2) + (6*x*y^2+6*y^2*z) + 3*y^4
sage: f.derivative(y, 5)
720*y
sage: f.derivative(z, 5)
0
sage: f.derivative(x, y, z)
12*y

sage: f = (1 + x + y^2 + z)^-1
sage: f.derivative(x)
-1 + (2*x+2*z) + (-3*x^2+2*y^2-6*x*z-3*z^2) + ... + O(x,y,z)^6
sage: f.derivative(y, 2)
-2 + (4*x+4*z) + (-6*x^2+12*y^2-12*x*z-6*z^2) + ... + O(x,y,z)^5
sage: f.derivative(x, y)
4*y + (-12*x*y-12*y*z) + (24*x^2*y-12*y^3+48*x*y*z+24*y*z^2)
 + (-40*x^3*y+48*x*y^3-120*x^2*y*z+48*y^3*z-120*x*y*z^2-40*y*z^3) + O(x,y,z)^5
sage: f.derivative(x, y, z)
(-12*y) + (48*x*y+48*y*z) + (-120*x^2*y+48*y^3-240*x*y*z-120*y*z^2) + O(x,y,z)^4

sage: R.<t> = QQ[]
sage: L.<x,y,z> = LazyPowerSeriesRing(R)
sage: f = ((t^2-3)*x + t*y^2 - t*z)^2
sage: f.derivative(t,x,t,y)
24*t*y
sage: f.derivative(t, 2)
((12*t^2-12)*x^2+(-12*t)*x*z+2*z^2) + (12*t*x*y^2+(-4)*y^2*z) + 2*y^4
sage: f.derivative(z, t)
((-6*t^2+6)*x+4*t*z) + ((-4*t)*y^2)
sage: f.derivative(t, 10)
0

sage: f = (1 + t*(x + y + z))^-1
sage: f.derivative(x, t, y)
4*t + ((-18*t^2)*x+(-18*t^2)*y+(-18*t^2)*z)
 + (48*t^3*x^2+96*t^3*x*y+48*t^3*y^2+96*t^3*x*z+96*t^3*y*z+48*t^3*z^2)
 + ... + O(x,y,z)^5
sage: f.derivative(t, 2)
(2*x^2+4*x*y+2*y^2+4*x*z+4*y*z+2*z^2) + ... + O(x,y,z)^7
sage: f.derivative(x, y, z, t)
(-18*t^2) + (96*t^3*x+96*t^3*y+96*t^3*z) + ... + O(x,y,z)^4
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> f = (x + y**Integer(2) + z)**Integer(3); f
(x^3+3*x^2*z+3*x*z^2+z^3) + (3*x^2*y^2+6*x*y^2*z+3*y^2*z^2) + (3*x*y^4+3*y^4*z) + y^6
>>> f.derivative(x)
(3*x^2+6*x*z+3*z^2) + (6*x*y^2+6*y^2*z) + 3*y^4
>>> f.derivative(y, Integer(5))
720*y
>>> f.derivative(z, Integer(5))
0
>>> f.derivative(x, y, z)
12*y

>>> f = (Integer(1) + x + y**Integer(2) + z)**-Integer(1)
>>> f.derivative(x)
-1 + (2*x+2*z) + (-3*x^2+2*y^2-6*x*z-3*z^2) + ... + O(x,y,z)^6
>>> f.derivative(y, Integer(2))
-2 + (4*x+4*z) + (-6*x^2+12*y^2-12*x*z-6*z^2) + ... + O(x,y,z)^5
>>> f.derivative(x, y)
4*y + (-12*x*y-12*y*z) + (24*x^2*y-12*y^3+48*x*y*z+24*y*z^2)
 + (-40*x^3*y+48*x*y^3-120*x^2*y*z+48*y^3*z-120*x*y*z^2-40*y*z^3) + O(x,y,z)^5
>>> f.derivative(x, y, z)
(-12*y) + (48*x*y+48*y*z) + (-120*x^2*y+48*y^3-240*x*y*z-120*y*z^2) + O(x,y,z)^4

>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> L = LazyPowerSeriesRing(R, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> f = ((t**Integer(2)-Integer(3))*x + t*y**Integer(2) - t*z)**Integer(2)
>>> f.derivative(t,x,t,y)
24*t*y
>>> f.derivative(t, Integer(2))
((12*t^2-12)*x^2+(-12*t)*x*z+2*z^2) + (12*t*x*y^2+(-4)*y^2*z) + 2*y^4
>>> f.derivative(z, t)
((-6*t^2+6)*x+4*t*z) + ((-4*t)*y^2)
>>> f.derivative(t, Integer(10))
0

>>> f = (Integer(1) + t*(x + y + z))**-Integer(1)
>>> f.derivative(x, t, y)
4*t + ((-18*t^2)*x+(-18*t^2)*y+(-18*t^2)*z)
 + (48*t^3*x^2+96*t^3*x*y+48*t^3*y^2+96*t^3*x*z+96*t^3*y*z+48*t^3*z^2)
 + ... + O(x,y,z)^5
>>> f.derivative(t, Integer(2))
(2*x^2+4*x*y+2*y^2+4*x*z+4*y*z+2*z^2) + ... + O(x,y,z)^7
>>> f.derivative(x, y, z, t)
(-18*t^2) + (96*t^3*x+96*t^3*y+96*t^3*z) + ... + O(x,y,z)^4
exponential()[source]

Return the exponential series of self.

This method is deprecated, use exp() instead.

integral(variable, constants=None)[source]

Return the integral of self with respect to variable.

INPUT:

  • variable – (optional) the variable to integrate

  • constants – (optional; keyword-only) list of integration constants for the integrals of self (the last constant corresponds to the first integral)

For multivariable series, only variable should be specified; the integration constant is taken to be \(0\).

Now we assume the series is univariate. If the first argument is a list, then this method iterprets it as integration constants. If it is a positive integer, the method interprets it as the number of times to integrate the function. If variable is not the variable of the power series, then the coefficients are integrated with respect to variable. If the integration constants are not specified, they are considered to be \(0\).

EXAMPLES:

sage: L.<t> = LazyPowerSeriesRing(QQ)
sage: f = 2 + 3*t + t^5
sage: f.integral()
2*t + 3/2*t^2 + 1/6*t^6
sage: f.integral([-2, -2])
-2 - 2*t + t^2 + 1/2*t^3 + 1/42*t^7
sage: f.integral(t)
2*t + 3/2*t^2 + 1/6*t^6
sage: f.integral(2)
t^2 + 1/2*t^3 + 1/42*t^7
sage: (t^3 + t^5).integral()
1/4*t^4 + 1/6*t^6
sage: L.zero().integral()
0
sage: L.zero().integral([0, 1, 2, 3])
t + t^2 + 1/2*t^3
sage: L([1, 2 ,3], constant=4).integral()
t + t^2 + t^3 + t^4 + 4/5*t^5 + 2/3*t^6 + O(t^7)
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('t',)); (t,) = L._first_ngens(1)
>>> f = Integer(2) + Integer(3)*t + t**Integer(5)
>>> f.integral()
2*t + 3/2*t^2 + 1/6*t^6
>>> f.integral([-Integer(2), -Integer(2)])
-2 - 2*t + t^2 + 1/2*t^3 + 1/42*t^7
>>> f.integral(t)
2*t + 3/2*t^2 + 1/6*t^6
>>> f.integral(Integer(2))
t^2 + 1/2*t^3 + 1/42*t^7
>>> (t**Integer(3) + t**Integer(5)).integral()
1/4*t^4 + 1/6*t^6
>>> L.zero().integral()
0
>>> L.zero().integral([Integer(0), Integer(1), Integer(2), Integer(3)])
t + t^2 + 1/2*t^3
>>> L([Integer(1), Integer(2) ,Integer(3)], constant=Integer(4)).integral()
t + t^2 + t^3 + t^4 + 4/5*t^5 + 2/3*t^6 + O(t^7)

We solve the ODE \(f'' - f' - 2 f = 0\) by solving for \(f''\), then integrating and applying a recursive definition:

sage: R.<C, D> = QQ[]
sage: L.<x> = LazyPowerSeriesRing(R)
sage: f = L.undefined()
sage: f.define((f.derivative() + 2*f).integral(constants=[C, D]))
sage: f
C + D*x + ((C+1/2*D)*x^2) + ((1/3*C+1/2*D)*x^3)
 + ((1/4*C+5/24*D)*x^4) + ((1/12*C+11/120*D)*x^5)
 + ((11/360*C+7/240*D)*x^6) + O(x^7)
sage: f.derivative(2) - f.derivative() - 2*f
O(x^7)
>>> from sage.all import *
>>> R = QQ['C, D']; (C, D,) = R._first_ngens(2)
>>> L = LazyPowerSeriesRing(R, names=('x',)); (x,) = L._first_ngens(1)
>>> f = L.undefined()
>>> f.define((f.derivative() + Integer(2)*f).integral(constants=[C, D]))
>>> f
C + D*x + ((C+1/2*D)*x^2) + ((1/3*C+1/2*D)*x^3)
 + ((1/4*C+5/24*D)*x^4) + ((1/12*C+11/120*D)*x^5)
 + ((11/360*C+7/240*D)*x^6) + O(x^7)
>>> f.derivative(Integer(2)) - f.derivative() - Integer(2)*f
O(x^7)

We compare this with the answer we get from the characteristic polynomial:

sage: g = C * exp(-x) + D * exp(2*x); g
(C+D) + ((-C+2*D)*x) + ((1/2*C+2*D)*x^2) + ((-1/6*C+4/3*D)*x^3)
 + ((1/24*C+2/3*D)*x^4) + ((-1/120*C+4/15*D)*x^5)
 + ((1/720*C+4/45*D)*x^6) + O(x^7)
sage: g.derivative(2) - g.derivative() - 2*g
O(x^7)
>>> from sage.all import *
>>> g = C * exp(-x) + D * exp(Integer(2)*x); g
(C+D) + ((-C+2*D)*x) + ((1/2*C+2*D)*x^2) + ((-1/6*C+4/3*D)*x^3)
 + ((1/24*C+2/3*D)*x^4) + ((-1/120*C+4/15*D)*x^5)
 + ((1/720*C+4/45*D)*x^6) + O(x^7)
>>> g.derivative(Integer(2)) - g.derivative() - Integer(2)*g
O(x^7)

Note that C and D are playing different roles, so we need to perform a substitution to the coefficients of f to recover the solution g:

sage: fp = f.map_coefficients(lambda c: c(C=C+D, D=2*D-C)); fp
(C+D) + ((-C+2*D)*x) + ((1/2*C+2*D)*x^2) + ((-1/6*C+4/3*D)*x^3)
 + ((1/24*C+2/3*D)*x^4) + ((-1/120*C+4/15*D)*x^5)
 + ((1/720*C+4/45*D)*x^6) + O(x^7)
sage: fp - g
O(x^7)
>>> from sage.all import *
>>> fp = f.map_coefficients(lambda c: c(C=C+D, D=Integer(2)*D-C)); fp
(C+D) + ((-C+2*D)*x) + ((1/2*C+2*D)*x^2) + ((-1/6*C+4/3*D)*x^3)
 + ((1/24*C+2/3*D)*x^4) + ((-1/120*C+4/15*D)*x^5)
 + ((1/720*C+4/45*D)*x^6) + O(x^7)
>>> fp - g
O(x^7)

We can integrate both the series and coefficients:

sage: R.<x,y,z> = QQ[]
sage: L.<t> = LazyPowerSeriesRing(R)
sage: f = (x*t^2 + y*t + z)^2; f
z^2 + 2*y*z*t + ((y^2+2*x*z)*t^2) + 2*x*y*t^3 + x^2*t^4
sage: f.integral(x)
x*z^2 + 2*x*y*z*t + ((x*y^2+x^2*z)*t^2) + x^2*y*t^3 + 1/3*x^3*t^4
sage: f.integral(t)
z^2*t + y*z*t^2 + ((1/3*y^2+2/3*x*z)*t^3) + 1/2*x*y*t^4 + 1/5*x^2*t^5
sage: f.integral(y, constants=[x*y*z])
x*y*z + y*z^2*t + 1/2*y^2*z*t^2 + ((1/9*y^3+2/3*x*y*z)*t^3) + 1/4*x*y^2*t^4 + 1/5*x^2*y*t^5
>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> L = LazyPowerSeriesRing(R, names=('t',)); (t,) = L._first_ngens(1)
>>> f = (x*t**Integer(2) + y*t + z)**Integer(2); f
z^2 + 2*y*z*t + ((y^2+2*x*z)*t^2) + 2*x*y*t^3 + x^2*t^4
>>> f.integral(x)
x*z^2 + 2*x*y*z*t + ((x*y^2+x^2*z)*t^2) + x^2*y*t^3 + 1/3*x^3*t^4
>>> f.integral(t)
z^2*t + y*z*t^2 + ((1/3*y^2+2/3*x*z)*t^3) + 1/2*x*y*t^4 + 1/5*x^2*t^5
>>> f.integral(y, constants=[x*y*z])
x*y*z + y*z^2*t + 1/2*y^2*z*t^2 + ((1/9*y^3+2/3*x*y*z)*t^3) + 1/4*x*y^2*t^4 + 1/5*x^2*y*t^5

We can integrate multivariate power series:

sage: R.<t> = QQ[]
sage: L.<x,y,z> = LazyPowerSeriesRing(R)
sage: f = ((t^2 + t) - t * y^2 + t^2 * (y + z))^2; f
(t^4+2*t^3+t^2) + ((2*t^4+2*t^3)*y+(2*t^4+2*t^3)*z)
 + ((t^4-2*t^3-2*t^2)*y^2+2*t^4*y*z+t^4*z^2)
 + ((-2*t^3)*y^3+(-2*t^3)*y^2*z) + t^2*y^4
sage: g = f.integral(x); g
((t^4+2*t^3+t^2)*x) + ((2*t^4+2*t^3)*x*y+(2*t^4+2*t^3)*x*z)
 + ((t^4-2*t^3-2*t^2)*x*y^2+2*t^4*x*y*z+t^4*x*z^2)
 + ((-2*t^3)*x*y^3+(-2*t^3)*x*y^2*z) + t^2*x*y^4
sage: g[0]
0
sage: g[1]
(t^4 + 2*t^3 + t^2)*x
sage: g[2]
(2*t^4 + 2*t^3)*x*y + (2*t^4 + 2*t^3)*x*z
sage: f.integral(z)
((t^4+2*t^3+t^2)*z) + ((2*t^4+2*t^3)*y*z+(t^4+t^3)*z^2)
 + ((t^4-2*t^3-2*t^2)*y^2*z+t^4*y*z^2+1/3*t^4*z^3)
 + ((-2*t^3)*y^3*z+(-t^3)*y^2*z^2) + t^2*y^4*z
sage: f.integral(t)
(1/5*t^5+1/2*t^4+1/3*t^3) + ((2/5*t^5+1/2*t^4)*y+(2/5*t^5+1/2*t^4)*z)
 + ((1/5*t^5-1/2*t^4-2/3*t^3)*y^2+2/5*t^5*y*z+1/5*t^5*z^2)
 + ((-1/2*t^4)*y^3+(-1/2*t^4)*y^2*z) + 1/3*t^3*y^4

sage: L.<x,y,z> = LazyPowerSeriesRing(QQ)
sage: (x + y - z^2).integral(z)
(x*z+y*z) + (-1/3*z^3)
>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> L = LazyPowerSeriesRing(R, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> f = ((t**Integer(2) + t) - t * y**Integer(2) + t**Integer(2) * (y + z))**Integer(2); f
(t^4+2*t^3+t^2) + ((2*t^4+2*t^3)*y+(2*t^4+2*t^3)*z)
 + ((t^4-2*t^3-2*t^2)*y^2+2*t^4*y*z+t^4*z^2)
 + ((-2*t^3)*y^3+(-2*t^3)*y^2*z) + t^2*y^4
>>> g = f.integral(x); g
((t^4+2*t^3+t^2)*x) + ((2*t^4+2*t^3)*x*y+(2*t^4+2*t^3)*x*z)
 + ((t^4-2*t^3-2*t^2)*x*y^2+2*t^4*x*y*z+t^4*x*z^2)
 + ((-2*t^3)*x*y^3+(-2*t^3)*x*y^2*z) + t^2*x*y^4
>>> g[Integer(0)]
0
>>> g[Integer(1)]
(t^4 + 2*t^3 + t^2)*x
>>> g[Integer(2)]
(2*t^4 + 2*t^3)*x*y + (2*t^4 + 2*t^3)*x*z
>>> f.integral(z)
((t^4+2*t^3+t^2)*z) + ((2*t^4+2*t^3)*y*z+(t^4+t^3)*z^2)
 + ((t^4-2*t^3-2*t^2)*y^2*z+t^4*y*z^2+1/3*t^4*z^3)
 + ((-2*t^3)*y^3*z+(-t^3)*y^2*z^2) + t^2*y^4*z
>>> f.integral(t)
(1/5*t^5+1/2*t^4+1/3*t^3) + ((2/5*t^5+1/2*t^4)*y+(2/5*t^5+1/2*t^4)*z)
 + ((1/5*t^5-1/2*t^4-2/3*t^3)*y^2+2/5*t^5*y*z+1/5*t^5*z^2)
 + ((-1/2*t^4)*y^3+(-1/2*t^4)*y^2*z) + 1/3*t^3*y^4

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3)
>>> (x + y - z**Integer(2)).integral(z)
(x*z+y*z) + (-1/3*z^3)
is_unit()[source]

Return whether this element is a unit in the ring.

EXAMPLES:

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: (2*z).is_unit()
False

sage: (1 + 2*z).is_unit()
True

sage: (3 + 2*z).is_unit()
False

sage: L.<x,y> = LazyPowerSeriesRing(ZZ)
sage: (-1 + 2*x + 3*x*y).is_unit()
True
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> (Integer(2)*z).is_unit()
False

>>> (Integer(1) + Integer(2)*z).is_unit()
True

>>> (Integer(3) + Integer(2)*z).is_unit()
False

>>> L = LazyPowerSeriesRing(ZZ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> (-Integer(1) + Integer(2)*x + Integer(3)*x*y).is_unit()
True
polynomial(degree=None, names=None)[source]

Return self as a polynomial if self is actually so.

INPUT:

  • degreeNone or an integer

  • names – names of the variables; if it is None, the name of the variables of the series is used

OUTPUT:

If degree is not None, the terms of the series of degree greater than degree are first truncated. If degree is None and the series is not a polynomial polynomial, a ValueError is raised.

EXAMPLES:

sage: L.<x,y> = LazyPowerSeriesRing(ZZ)
sage: f = x^2 + y*x - x + 2; f
2 + (-x) + (x^2+x*y)
sage: f.polynomial()
x^2 + x*y - x + 2
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> f = x**Integer(2) + y*x - x + Integer(2); f
2 + (-x) + (x^2+x*y)
>>> f.polynomial()
x^2 + x*y - x + 2
revert()[source]

Return the compositional inverse of self.

Given a Taylor series \(f\) in one variable, the compositional inverse is a power series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

  • \(\mathrm{val}(f) = 1\), or

  • \(f = a + b z\) with \(a, b \neq 0\).

EXAMPLES:

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: (2*z).revert()
1/2*z
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

sage: s = L(degree=1, constant=-1)
sage: s.revert()
-z - z^2 - z^3 + O(z^4)

sage: s = L(degree=1, constant=1)
sage: s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> (Integer(2)*z).revert()
1/2*z
>>> (z-z**Integer(2)).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

>>> s = L(degree=Integer(1), constant=-Integer(1))
>>> s.revert()
-z - z^2 - z^3 + O(z^4)

>>> s = L(degree=Integer(1), constant=Integer(1))
>>> s.revert()
z - z^2 + z^3 - z^4 + z^5 - z^6 + z^7 + O(z^8)

Warning

For series not known to be eventually constant (e.g., being defined by a function) with approximate valuation \(\leq 1\) (but not necessarily its true valuation), this assumes that this is the actual valuation:

sage: f = L(lambda n: n if n > 2 else 0)
sage: f.revert()
<repr(... failed: ValueError: generator already executing>
>>> from sage.all import *
>>> f = L(lambda n: n if n > Integer(2) else Integer(0))
>>> f.revert()
<repr(... failed: ValueError: generator already executing>
class sage.rings.lazy_series.LazyPowerSeries_gcd_mixin[source]

Bases: object

A lazy power series that also implements the GCD algorithm.

gcd(other)[source]

Return the greatest common divisor of self and other.

EXAMPLES:

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: a = 16*x^5 / (1 - 5*x)
sage: b = (22*x^2 + x^8) / (1 - 4*x^2)
sage: a.gcd(b)
x^2
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x',)); (x,) = L._first_ngens(1)
>>> a = Integer(16)*x**Integer(5) / (Integer(1) - Integer(5)*x)
>>> b = (Integer(22)*x**Integer(2) + x**Integer(8)) / (Integer(1) - Integer(4)*x**Integer(2))
>>> a.gcd(b)
x^2
xgcd(f)[source]

Return the extended gcd of self and f.

OUTPUT:

A triple (g, s, t) such that g is the gcd of self and f, and s and t are cofactors satisfying the Bezout identity

\[g = s \cdot \mathrm{self} + t \cdot f.\]

EXAMPLES:

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: a = 16*x^5 / (1 - 2*x)
sage: b = (22*x^3 + x^8) / (1 - 3*x^2)
sage: g, s, t = a.xgcd(b)
sage: g
x^3
sage: s
1/22 - 41/242*x^2 - 8/121*x^3 + 120/1331*x^4 + 1205/5324*x^5 + 316/14641*x^6 + O(x^7)
sage: t
1/22 - 41/242*x^2 - 8/121*x^3 + 120/1331*x^4 + 1205/5324*x^5 + 316/14641*x^6 + O(x^7)

sage: LazyPowerSeriesRing.options.halting_precision(20)  # verify up to degree 20

sage: g == s * a + t * b
True

sage: a = 16*x^5 / (1 - 2*x)
sage: b = (-16*x^5 + x^8) / (1 - 3*x^2)
sage: g, s, t = a.xgcd(b)
sage: g
x^5
sage: s
1/16 - 1/16*x - 3/16*x^2 + 1/8*x^3 - 17/256*x^4 + 9/128*x^5 + 1/128*x^6 + O(x^7)
sage: t
1/16*x - 1/16*x^2 - 3/16*x^3 + 1/8*x^4 - 17/256*x^5 + 9/128*x^6 + 1/128*x^7 + O(x^8)
sage: g == s * a + t * b
True

sage: # needs sage.rings.finite_rings
sage: L.<x> = LazyPowerSeriesRing(GF(2))
sage: a = L(lambda n: n % 2, valuation=3); a
x^3 + x^5 + x^7 + x^9 + O(x^10)
sage: b = L(lambda n: binomial(n,2) % 2, valuation=3); b
x^3 + x^6 + x^7 + O(x^10)
sage: g, s, t = a.xgcd(b)
sage: g
x^3
sage: s
1 + x + x^3 + x^4 + x^5 + O(x^7)
sage: t
x + x^2 + x^4 + x^5 + x^6 + O(x^8)
sage: g == s * a + t * b
True

sage: LazyPowerSeriesRing.options._reset()  # reset the options
>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x',)); (x,) = L._first_ngens(1)
>>> a = Integer(16)*x**Integer(5) / (Integer(1) - Integer(2)*x)
>>> b = (Integer(22)*x**Integer(3) + x**Integer(8)) / (Integer(1) - Integer(3)*x**Integer(2))
>>> g, s, t = a.xgcd(b)
>>> g
x^3
>>> s
1/22 - 41/242*x^2 - 8/121*x^3 + 120/1331*x^4 + 1205/5324*x^5 + 316/14641*x^6 + O(x^7)
>>> t
1/22 - 41/242*x^2 - 8/121*x^3 + 120/1331*x^4 + 1205/5324*x^5 + 316/14641*x^6 + O(x^7)

>>> LazyPowerSeriesRing.options.halting_precision(Integer(20))  # verify up to degree 20

>>> g == s * a + t * b
True

>>> a = Integer(16)*x**Integer(5) / (Integer(1) - Integer(2)*x)
>>> b = (-Integer(16)*x**Integer(5) + x**Integer(8)) / (Integer(1) - Integer(3)*x**Integer(2))
>>> g, s, t = a.xgcd(b)
>>> g
x^5
>>> s
1/16 - 1/16*x - 3/16*x^2 + 1/8*x^3 - 17/256*x^4 + 9/128*x^5 + 1/128*x^6 + O(x^7)
>>> t
1/16*x - 1/16*x^2 - 3/16*x^3 + 1/8*x^4 - 17/256*x^5 + 9/128*x^6 + 1/128*x^7 + O(x^8)
>>> g == s * a + t * b
True

>>> # needs sage.rings.finite_rings
>>> L = LazyPowerSeriesRing(GF(Integer(2)), names=('x',)); (x,) = L._first_ngens(1)
>>> a = L(lambda n: n % Integer(2), valuation=Integer(3)); a
x^3 + x^5 + x^7 + x^9 + O(x^10)
>>> b = L(lambda n: binomial(n,Integer(2)) % Integer(2), valuation=Integer(3)); b
x^3 + x^6 + x^7 + O(x^10)
>>> g, s, t = a.xgcd(b)
>>> g
x^3
>>> s
1 + x + x^3 + x^4 + x^5 + O(x^7)
>>> t
x + x^2 + x^4 + x^5 + x^6 + O(x^8)
>>> g == s * a + t * b
True

>>> LazyPowerSeriesRing.options._reset()  # reset the options
class sage.rings.lazy_series.LazySymmetricFunction(parent, coeff_stream)[source]

Bases: LazyCompletionGradedAlgebraElement

A symmetric function where each degree is computed lazily.

EXAMPLES:

sage: s = SymmetricFunctions(ZZ).s()                                            # needs sage.modules
sage: L = LazySymmetricFunctions(s)                                             # needs sage.modules
>>> from sage.all import *
>>> s = SymmetricFunctions(ZZ).s()                                            # needs sage.modules
>>> L = LazySymmetricFunctions(s)                                             # needs sage.modules
arithmetic_product(*args)[source]

Return the arithmetic product of self with g.

The arithmetic product is a binary operation \(\boxdot\) on the ring of symmetric functions which is bilinear in its two arguments and satisfies

\[p_{\lambda} \boxdot p_{\mu} = \prod\limits_{i \geq 1, j \geq 1} p_{\mathrm{lcm}(\lambda_i, \mu_j)}^{\mathrm{gcd}(\lambda_i, \mu_j)}\]

for any two partitions \(\lambda = (\lambda_1, \lambda_2, \lambda_3, \dots )\) and \(\mu = (\mu_1, \mu_2, \mu_3, \dots )\) (where \(p_{\nu}\) denotes the power-sum symmetric function indexed by the partition \(\nu\), and \(p_i\) denotes the \(i\)-th power-sum symmetric function). This is enough to define the arithmetic product if the base ring is torsion-free as a \(\ZZ\)-module; for all other cases the arithmetic product is uniquely determined by requiring it to be functorial in the base ring. See http://mathoverflow.net/questions/138148/ for a discussion of this arithmetic product.

Warning

The operation \(f \boxdot g\) was originally defined only for symmetric functions \(f\) and \(g\) without constant term. We extend this definition using the convention that the least common multiple of any integer with \(0\) is \(0\).

If \(f\) and \(g\) are two symmetric functions which are homogeneous of degrees \(a\) and \(b\), respectively, then \(f \boxdot g\) is homogeneous of degree \(ab\).

The arithmetic product is commutative and associative and has unity \(e_1 = p_1 = h_1\).

For species \(M\) and \(N\) such that \(M[\varnothing] = N[\varnothing] = \varnothing\), their arithmetic product is the species \(M \boxdot N\) of “\(M\)-assemblies of cloned \(N\)-structures”. This operation is defined and several examples are given in [MM2008].

INPUT:

  • g – a cycle index series having the same parent as self

OUTPUT: the arithmetic product of self with g

EXAMPLES:

For \(C\) the species of (oriented) cycles and \(L_{+}\) the species of nonempty linear orders, \(C \boxdot L_{+}\) corresponds to the species of “regular octopuses”; a \((C \boxdot L_{+})\)-structure is a cycle of some length, each of whose elements is an ordered list of a length which is consistent for all the lists in the structure.

sage: R.<q> = QQ[]
sage: p = SymmetricFunctions(R).p()                                         # needs sage.modules
sage: m = SymmetricFunctions(R).m()                                         # needs sage.modules
sage: L = LazySymmetricFunctions(m)                                         # needs sage.modules

sage: # needs sage.modules
sage: C = species.CycleSpecies().cycle_index_series()
sage: c = L(lambda n: C[n])
sage: Lplus = L(lambda n: p([1]*n), valuation=1)
sage: r = c.arithmetic_product(Lplus); r                                    # needs sage.libs.pari
m[1] + (3*m[1,1]+2*m[2])
 + (8*m[1,1,1]+4*m[2,1]+2*m[3])
 + (42*m[1,1,1,1]+21*m[2,1,1]+12*m[2,2]+7*m[3,1]+3*m[4])
 + (144*m[1,1,1,1,1]+72*m[2,1,1,1]+36*m[2,2,1]+24*m[3,1,1]+12*m[3,2]+6*m[4,1]+2*m[5])
 + ...
 + O^7
>>> from sage.all import *
>>> R = QQ['q']; (q,) = R._first_ngens(1)
>>> p = SymmetricFunctions(R).p()                                         # needs sage.modules
>>> m = SymmetricFunctions(R).m()                                         # needs sage.modules
>>> L = LazySymmetricFunctions(m)                                         # needs sage.modules

>>> # needs sage.modules
>>> C = species.CycleSpecies().cycle_index_series()
>>> c = L(lambda n: C[n])
>>> Lplus = L(lambda n: p([Integer(1)]*n), valuation=Integer(1))
>>> r = c.arithmetic_product(Lplus); r                                    # needs sage.libs.pari
m[1] + (3*m[1,1]+2*m[2])
 + (8*m[1,1,1]+4*m[2,1]+2*m[3])
 + (42*m[1,1,1,1]+21*m[2,1,1]+12*m[2,2]+7*m[3,1]+3*m[4])
 + (144*m[1,1,1,1,1]+72*m[2,1,1,1]+36*m[2,2,1]+24*m[3,1,1]+12*m[3,2]+6*m[4,1]+2*m[5])
 + ...
 + O^7

In particular, the number of regular octopuses is:

sage: [r[n].coefficient([1]*n) for n in range(8)]                           # needs sage.libs.pari sage.modules
[0, 1, 3, 8, 42, 144, 1440, 5760]
>>> from sage.all import *
>>> [r[n].coefficient([Integer(1)]*n) for n in range(Integer(8))]                           # needs sage.libs.pari sage.modules
[0, 1, 3, 8, 42, 144, 1440, 5760]

It is shown in [MM2008] that the exponential generating function for regular octopuses satisfies \((C \boxdot L_{+}) (x) = \sum_{n \geq 1} \sigma (n) (n - 1)! \frac{x^{n}}{n!}\) (where \(\sigma (n)\) is the sum of the divisors of \(n\)).

sage: [sum(divisors(i))*factorial(i-1) for i in range(1,8)]                 # needs sage.modules
[1, 3, 8, 42, 144, 1440, 5760]
>>> from sage.all import *
>>> [sum(divisors(i))*factorial(i-Integer(1)) for i in range(Integer(1),Integer(8))]                 # needs sage.modules
[1, 3, 8, 42, 144, 1440, 5760]

AUTHORS:

  • Andrew Gainer-Dewar (2013)

REFERENCES:

compositional_inverse()[source]

Return the compositional inverse of self.

Given a symmetric function \(f\), the compositional inverse is a symmetric function \(g\) over the same base ring, such that \(f \circ g = p_1\). Thus, it is the inverse with respect to plethystic substitution.

The compositional inverse exists if and only if:

  • \(\mathrm{val}(f) = 1\), or

  • \(f = a + b p_1\) with \(a, b \neq 0\).

EXAMPLES:

sage: # needs sage.modules
sage: h = SymmetricFunctions(QQ).h()
sage: L = LazySymmetricFunctions(h)
sage: f = L(lambda n: h[n]) - 1
sage: f(f.revert())
h[1] + O^8
>>> from sage.all import *
>>> # needs sage.modules
>>> h = SymmetricFunctions(QQ).h()
>>> L = LazySymmetricFunctions(h)
>>> f = L(lambda n: h[n]) - Integer(1)
>>> f(f.revert())
h[1] + O^8

ALGORITHM:

Let \(F\) be a symmetric function with valuation \(1\), i.e., whose constant term vanishes and whose degree one term equals \(b p_1\). Then

\[(F - b p_1) \circ G = F \circ G - b p_1 \circ G = p_1 - b G,\]

and therefore \(G = (p_1 - (F - b p_1) \circ G) / b\), which allows recursive computation of \(G\).

See also

The compositional inverse \(\Omega\) of the symmetric function \(h_1 + h_2 + \dots\) can be handled much more efficiently using specialized methods. See LogarithmCycleIndexSeries()

AUTHORS:

  • Andrew Gainer-Dewar

  • Martin Rubey

derivative_with_respect_to_p1(n=1)[source]

Return the symmetric function obtained by taking the derivative of self with respect to the power-sum symmetric function \(p_1\) when the expansion of self in the power-sum basis is considered as a polynomial in \(p_k\)’s (with \(k \geq 1\)).

This is the same as skewing self by the first power-sum symmetric function \(p_1\).

INPUT:

  • n – (default: 1) nonnegative integer which determines which power of the derivative is taken

EXAMPLES:

The species \(E\) of sets satisfies the relationship \(E' = E\):

sage: # needs sage.modules
sage: h = SymmetricFunctions(QQ).h()
sage: T = LazySymmetricFunctions(h)
sage: E = T(lambda n: h[n])
sage: E - E.derivative_with_respect_to_p1()
O^6
>>> from sage.all import *
>>> # needs sage.modules
>>> h = SymmetricFunctions(QQ).h()
>>> T = LazySymmetricFunctions(h)
>>> E = T(lambda n: h[n])
>>> E - E.derivative_with_respect_to_p1()
O^6

The species \(C\) of cyclic orderings and the species \(L\) of linear orderings satisfy the relationship \(C' = L\):

sage: # needs sage.modules
sage: p = SymmetricFunctions(QQ).p()
sage: C = T(lambda n: (sum(euler_phi(k)*p([k])**(n//k)
....:                      for k in divisors(n))/n if n > 0 else 0))
sage: L = T(lambda n: p([1]*n))
sage: L - C.derivative_with_respect_to_p1()                                 # needs sage.libs.pari
O^6
>>> from sage.all import *
>>> # needs sage.modules
>>> p = SymmetricFunctions(QQ).p()
>>> C = T(lambda n: (sum(euler_phi(k)*p([k])**(n//k)
...                      for k in divisors(n))/n if n > Integer(0) else Integer(0)))
>>> L = T(lambda n: p([Integer(1)]*n))
>>> L - C.derivative_with_respect_to_p1()                                 # needs sage.libs.pari
O^6
functorial_composition(*args)[source]

Return the functorial composition of self and g.

Let \(X\) be a finite set of cardinality \(m\). For a group action of the symmetric group \(g: S_n \to S_X\) and a (possibly virtual) representation of the symmetric group on \(X\), \(f: S_X \to GL(V)\), the functorial composition is the (virtual) representation of the symmetric group \(f \Box g: S_n \to GL(V)\) given by \(\sigma \mapsto f(g(\sigma))\).

This is more naturally phrased in the language of combinatorial species. Let \(F\) and \(G\) be species, then their functorial composition is the species \(F \Box G\) with \((F \Box G) [A] = F[ G[A] ]\). In other words, an \((F \Box G)\)-structure on a set \(A\) of labels is an \(F\)-structure whose labels are the set of all \(G\)-structures on \(A\).

The Frobenius character (or cycle index series) of \(F \Box G\) can be computed as follows, see section 2.2 of [BLL1998]):

\[\sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_{n}} \operatorname{fix} F[ (G[\sigma])_{1}, (G[\sigma])_{2}, \ldots ] \, p_{1}^{\sigma_{1}} p_{2}^{\sigma_{2}} \cdots.\]

Warning

The operation \(f \Box g\) only makes sense when \(g\) corresponds to a permutation representation, i.e., a group action.

EXAMPLES:

The species \(G\) of simple graphs can be expressed in terms of a functorial composition: \(G = \mathfrak{p} \Box \mathfrak{p}_{2}\), where \(\mathfrak{p}\) is the SubsetSpecies.:

sage: # needs sage.modules
sage: R.<q> = QQ[]
sage: h = SymmetricFunctions(R).h()
sage: m = SymmetricFunctions(R).m()
sage: L = LazySymmetricFunctions(m)
sage: P = L(lambda n: sum(q^k*h[n-k]*h[k] for k in range(n+1)))
sage: P2 = L(lambda n: h[2]*h[n-2], valuation=2)
sage: P.functorial_composition(P2)[:4]                                      # needs sage.libs.pari
[m[],
 m[1],
 (q+1)*m[1, 1] + (q+1)*m[2],
 (q^3+3*q^2+3*q+1)*m[1, 1, 1] + (q^3+2*q^2+2*q+1)*m[2, 1] + (q^3+q^2+q+1)*m[3]]
>>> from sage.all import *
>>> # needs sage.modules
>>> R = QQ['q']; (q,) = R._first_ngens(1)
>>> h = SymmetricFunctions(R).h()
>>> m = SymmetricFunctions(R).m()
>>> L = LazySymmetricFunctions(m)
>>> P = L(lambda n: sum(q**k*h[n-k]*h[k] for k in range(n+Integer(1))))
>>> P2 = L(lambda n: h[Integer(2)]*h[n-Integer(2)], valuation=Integer(2))
>>> P.functorial_composition(P2)[:Integer(4)]                                      # needs sage.libs.pari
[m[],
 m[1],
 (q+1)*m[1, 1] + (q+1)*m[2],
 (q^3+3*q^2+3*q+1)*m[1, 1, 1] + (q^3+2*q^2+2*q+1)*m[2, 1] + (q^3+q^2+q+1)*m[3]]

For example, there are:

sage: P.functorial_composition(P2)[4].coefficient([4])[3]                   # needs sage.libs.pari sage.modules
3
>>> from sage.all import *
>>> P.functorial_composition(P2)[Integer(4)].coefficient([Integer(4)])[Integer(3)]                   # needs sage.libs.pari sage.modules
3

unlabelled graphs on 4 vertices and 3 edges, and:

sage: P.functorial_composition(P2)[4].coefficient([2,2])[3]                 # needs sage.libs.pari sage.modules
8
>>> from sage.all import *
>>> P.functorial_composition(P2)[Integer(4)].coefficient([Integer(2),Integer(2)])[Integer(3)]                 # needs sage.libs.pari sage.modules
8

labellings of their vertices with two 1s and two 2s.

The symmetric function \(h_1 \sum_n h_n\) is the neutral element with respect to functorial composition:

sage: # needs sage.modules
sage: p = SymmetricFunctions(QQ).p()
sage: h = SymmetricFunctions(QQ).h()
sage: e = SymmetricFunctions(QQ).e()
sage: L = LazySymmetricFunctions(h)
sage: H = L(lambda n: h[n])
sage: Ep = p[1]*H.derivative_with_respect_to_p1(); Ep
h[1] + (h[1,1]) + (h[2,1]) + (h[3,1]) + (h[4,1]) + (h[5,1]) + O^7
sage: f = L(lambda n: h[n-n//2, n//2])
sage: f - Ep.functorial_composition(f)                                      # needs sage.libs.pari
O^7
>>> from sage.all import *
>>> # needs sage.modules
>>> p = SymmetricFunctions(QQ).p()
>>> h = SymmetricFunctions(QQ).h()
>>> e = SymmetricFunctions(QQ).e()
>>> L = LazySymmetricFunctions(h)
>>> H = L(lambda n: h[n])
>>> Ep = p[Integer(1)]*H.derivative_with_respect_to_p1(); Ep
h[1] + (h[1,1]) + (h[2,1]) + (h[3,1]) + (h[4,1]) + (h[5,1]) + O^7
>>> f = L(lambda n: h[n-n//Integer(2), n//Integer(2)])
>>> f - Ep.functorial_composition(f)                                      # needs sage.libs.pari
O^7

The symmetric function \(\sum_n h_n\) is a left absorbing element:

sage: # needs sage.modules
sage: H.functorial_composition(f) - H
O^7
>>> from sage.all import *
>>> # needs sage.modules
>>> H.functorial_composition(f) - H
O^7

The functorial composition distributes over the sum:

sage: # needs sage.modules
sage: F1 = L(lambda n: h[n])
sage: F2 = L(lambda n: e[n])
sage: f1 = F1.functorial_composition(f)
sage: f2 = F2.functorial_composition(f)
sage: (F1 + F2).functorial_composition(f) - f1 - f2         # long time
O^7
>>> from sage.all import *
>>> # needs sage.modules
>>> F1 = L(lambda n: h[n])
>>> F2 = L(lambda n: e[n])
>>> f1 = F1.functorial_composition(f)
>>> f2 = F2.functorial_composition(f)
>>> (F1 + F2).functorial_composition(f) - f1 - f2         # long time
O^7
is_unit()[source]

Return whether this element is a unit in the ring.

EXAMPLES:

sage: # needs sage.modules
sage: m = SymmetricFunctions(ZZ).m()
sage: L = LazySymmetricFunctions(m)
sage: L(2*m[1]).is_unit()
False
sage: L(-1 + 2*m[1]).is_unit()
True
sage: L(2 + m[1]).is_unit()
False
sage: m = SymmetricFunctions(QQ).m()
sage: L = LazySymmetricFunctions(m)
sage: L(2 + 3*m[1]).is_unit()
True
>>> from sage.all import *
>>> # needs sage.modules
>>> m = SymmetricFunctions(ZZ).m()
>>> L = LazySymmetricFunctions(m)
>>> L(Integer(2)*m[Integer(1)]).is_unit()
False
>>> L(-Integer(1) + Integer(2)*m[Integer(1)]).is_unit()
True
>>> L(Integer(2) + m[Integer(1)]).is_unit()
False
>>> m = SymmetricFunctions(QQ).m()
>>> L = LazySymmetricFunctions(m)
>>> L(Integer(2) + Integer(3)*m[Integer(1)]).is_unit()
True
legendre_transform()[source]

Return the Legendre transform of self.

Given a symmetric function \(f\) of valuation 2, the Legendre transform of \(f\) is the unique symmetric function \(g\) of valuation 2 over the same base ring, such that

\[g \circ \partial_{p_1} f + f = p_1 \partial_{p_1} f.\]

This implies that the derivatives of \(f\) and \(g\) with respect to \(p_1\) are inverses of each other with respect to plethystic substitution.

The Legendre transform is an involution.

See also

revert()

EXAMPLES:

sage: p = SymmetricFunctions(QQ).p()
sage: s = SymmetricFunctions(QQ).s()
sage: lp = LazySymmetricFunctions(p)
sage: A = lp(s([2]))
sage: A.legendre_transform()
(1/2*p[1,1]-1/2*p[2]) + O^9

sage: def asso(n):
....:     return p.sum_of_terms((Partition([d] * (n // d)),
....:          euler_phi(d) / n) for d in divisors(n))

sage: A = lp(asso, valuation=2)
sage: A.legendre_transform()[:5]
[1/2*p[1, 1] - 1/2*p[2],
-1/3*p[1, 1, 1] - 2/3*p[3],
1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] - 1/2*p[4]]
>>> from sage.all import *
>>> p = SymmetricFunctions(QQ).p()
>>> s = SymmetricFunctions(QQ).s()
>>> lp = LazySymmetricFunctions(p)
>>> A = lp(s([Integer(2)]))
>>> A.legendre_transform()
(1/2*p[1,1]-1/2*p[2]) + O^9

>>> def asso(n):
...     return p.sum_of_terms((Partition([d] * (n // d)),
...          euler_phi(d) / n) for d in divisors(n))

>>> A = lp(asso, valuation=Integer(2))
>>> A.legendre_transform()[:Integer(5)]
[1/2*p[1, 1] - 1/2*p[2],
-1/3*p[1, 1, 1] - 2/3*p[3],
1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] - 1/2*p[4]]
plethysm(*args)[source]

Return the composition of self with g.

The arity of self must be equal to the number of arguments provided.

Given a lazy symmetric function \(f\) of arity \(n\) and a tuple of lazy symmetric functions \(g = (g_1,\dots, g_n)\) over the same base ring, the composition (or plethysm) \((f \circ g)\) is defined if and only if for each \(1\leq i\leq n\):

  • \(g_i = 0\), or

  • setting all alphabets except the \(i\)-th in \(f\) to zero yields a symmetric function with only finitely many nonzero coefficients, or

  • \(\mathrm{val}(g) > 0\).

If \(f\) is a univariate ‘exact’ lazy symmetric function, we can check whether \(f\) has only finitely many nonzero coefficients. However, if \(f\) has larger arity, we have no way to test whether setting all but one alphabets of \(f\) to zero yields a polynomial, except if \(f\) itself is ‘exact’ and therefore a symmetric function with only finitely many nonzero coefficients.

INPUT:

  • g – other (lazy) symmetric functions

Todo

Allow specification of degree one elements.

EXAMPLES:

sage: # needs sage.modules
sage: P.<q> = QQ[]
sage: s = SymmetricFunctions(P).s()
sage: L = LazySymmetricFunctions(s)
sage: f = s[2]
sage: g = s[3]
sage: L(f)(L(g)) - L(f(g))
0
sage: f = s[2] + s[2,1]
sage: g = s[1] + s[2,2]
sage: L(f)(L(g)) - L(f(g))
0
sage: L(f)(g) - L(f(g))
0
sage: f = s[2] + s[2,1]
sage: g = s[1] + s[2,2]
sage: L(f)(L(q*g)) - L(f(q*g))
0
>>> from sage.all import *
>>> # needs sage.modules
>>> P = QQ['q']; (q,) = P._first_ngens(1)
>>> s = SymmetricFunctions(P).s()
>>> L = LazySymmetricFunctions(s)
>>> f = s[Integer(2)]
>>> g = s[Integer(3)]
>>> L(f)(L(g)) - L(f(g))
0
>>> f = s[Integer(2)] + s[Integer(2),Integer(1)]
>>> g = s[Integer(1)] + s[Integer(2),Integer(2)]
>>> L(f)(L(g)) - L(f(g))
0
>>> L(f)(g) - L(f(g))
0
>>> f = s[Integer(2)] + s[Integer(2),Integer(1)]
>>> g = s[Integer(1)] + s[Integer(2),Integer(2)]
>>> L(f)(L(q*g)) - L(f(q*g))
0

The Frobenius character of the permutation action on set partitions is a plethysm:

sage: # needs sage.modules
sage: s = SymmetricFunctions(QQ).s()
sage: S = LazySymmetricFunctions(s)
sage: E1 = S(lambda n: s[n], valuation=1)
sage: E = 1 + E1
sage: P = E(E1)
sage: P[:5]
[s[], s[1], 2*s[2], s[2, 1] + 3*s[3], 2*s[2, 2] + 2*s[3, 1] + 5*s[4]]
>>> from sage.all import *
>>> # needs sage.modules
>>> s = SymmetricFunctions(QQ).s()
>>> S = LazySymmetricFunctions(s)
>>> E1 = S(lambda n: s[n], valuation=Integer(1))
>>> E = Integer(1) + E1
>>> P = E(E1)
>>> P[:Integer(5)]
[s[], s[1], 2*s[2], s[2, 1] + 3*s[3], 2*s[2, 2] + 2*s[3, 1] + 5*s[4]]

The plethysm with a tensor product is also implemented:

sage: # needs sage.modules
sage: s = SymmetricFunctions(QQ).s()
sage: X = tensor([s[1],s[[]]])
sage: Y = tensor([s[[]],s[1]])
sage: S = LazySymmetricFunctions(s)
sage: S2 = LazySymmetricFunctions(tensor([s, s]))
sage: A = S(s[1,1,1])
sage: B = S2(X+Y)
sage: A(B)                                                                  # needs lrcalc_python
(s[]#s[1,1,1]+s[1]#s[1,1]+s[1,1]#s[1]+s[1,1,1]#s[])

sage: H = S(lambda n: s[n])                                                 # needs sage.modules
sage: H(S2(X*Y))                                                            # needs lrcalc_python sage.modules
(s[]#s[]) + (s[1]#s[1]) + (s[1,1]#s[1,1]+s[2]#s[2])
 + (s[1,1,1]#s[1,1,1]+s[2,1]#s[2,1]+s[3]#s[3]) + O^7
sage: H(S2(X+Y))                                                            # needs sage.modules
(s[]#s[]) + (s[]#s[1]+s[1]#s[]) + (s[]#s[2]+s[1]#s[1]+s[2]#s[])
 + (s[]#s[3]+s[1]#s[2]+s[2]#s[1]+s[3]#s[])
 + (s[]#s[4]+s[1]#s[3]+s[2]#s[2]+s[3]#s[1]+s[4]#s[])
 + (s[]#s[5]+s[1]#s[4]+s[2]#s[3]+s[3]#s[2]+s[4]#s[1]+s[5]#s[])
 + (s[]#s[6]+s[1]#s[5]+s[2]#s[4]+s[3]#s[3]+s[4]#s[2]+s[5]#s[1]+s[6]#s[])
 + O^7
>>> from sage.all import *
>>> # needs sage.modules
>>> s = SymmetricFunctions(QQ).s()
>>> X = tensor([s[Integer(1)],s[[]]])
>>> Y = tensor([s[[]],s[Integer(1)]])
>>> S = LazySymmetricFunctions(s)
>>> S2 = LazySymmetricFunctions(tensor([s, s]))
>>> A = S(s[Integer(1),Integer(1),Integer(1)])
>>> B = S2(X+Y)
>>> A(B)                                                                  # needs lrcalc_python
(s[]#s[1,1,1]+s[1]#s[1,1]+s[1,1]#s[1]+s[1,1,1]#s[])

>>> H = S(lambda n: s[n])                                                 # needs sage.modules
>>> H(S2(X*Y))                                                            # needs lrcalc_python sage.modules
(s[]#s[]) + (s[1]#s[1]) + (s[1,1]#s[1,1]+s[2]#s[2])
 + (s[1,1,1]#s[1,1,1]+s[2,1]#s[2,1]+s[3]#s[3]) + O^7
>>> H(S2(X+Y))                                                            # needs sage.modules
(s[]#s[]) + (s[]#s[1]+s[1]#s[]) + (s[]#s[2]+s[1]#s[1]+s[2]#s[])
 + (s[]#s[3]+s[1]#s[2]+s[2]#s[1]+s[3]#s[])
 + (s[]#s[4]+s[1]#s[3]+s[2]#s[2]+s[3]#s[1]+s[4]#s[])
 + (s[]#s[5]+s[1]#s[4]+s[2]#s[3]+s[3]#s[2]+s[4]#s[1]+s[5]#s[])
 + (s[]#s[6]+s[1]#s[5]+s[2]#s[4]+s[3]#s[3]+s[4]#s[2]+s[5]#s[1]+s[6]#s[])
 + O^7
plethystic_inverse()[source]

Return the compositional inverse of self.

Given a symmetric function \(f\), the compositional inverse is a symmetric function \(g\) over the same base ring, such that \(f \circ g = p_1\). Thus, it is the inverse with respect to plethystic substitution.

The compositional inverse exists if and only if:

  • \(\mathrm{val}(f) = 1\), or

  • \(f = a + b p_1\) with \(a, b \neq 0\).

EXAMPLES:

sage: # needs sage.modules
sage: h = SymmetricFunctions(QQ).h()
sage: L = LazySymmetricFunctions(h)
sage: f = L(lambda n: h[n]) - 1
sage: f(f.revert())
h[1] + O^8
>>> from sage.all import *
>>> # needs sage.modules
>>> h = SymmetricFunctions(QQ).h()
>>> L = LazySymmetricFunctions(h)
>>> f = L(lambda n: h[n]) - Integer(1)
>>> f(f.revert())
h[1] + O^8

ALGORITHM:

Let \(F\) be a symmetric function with valuation \(1\), i.e., whose constant term vanishes and whose degree one term equals \(b p_1\). Then

\[(F - b p_1) \circ G = F \circ G - b p_1 \circ G = p_1 - b G,\]

and therefore \(G = (p_1 - (F - b p_1) \circ G) / b\), which allows recursive computation of \(G\).

See also

The compositional inverse \(\Omega\) of the symmetric function \(h_1 + h_2 + \dots\) can be handled much more efficiently using specialized methods. See LogarithmCycleIndexSeries()

AUTHORS:

  • Andrew Gainer-Dewar

  • Martin Rubey

revert()[source]

Return the compositional inverse of self.

Given a symmetric function \(f\), the compositional inverse is a symmetric function \(g\) over the same base ring, such that \(f \circ g = p_1\). Thus, it is the inverse with respect to plethystic substitution.

The compositional inverse exists if and only if:

  • \(\mathrm{val}(f) = 1\), or

  • \(f = a + b p_1\) with \(a, b \neq 0\).

EXAMPLES:

sage: # needs sage.modules
sage: h = SymmetricFunctions(QQ).h()
sage: L = LazySymmetricFunctions(h)
sage: f = L(lambda n: h[n]) - 1
sage: f(f.revert())
h[1] + O^8
>>> from sage.all import *
>>> # needs sage.modules
>>> h = SymmetricFunctions(QQ).h()
>>> L = LazySymmetricFunctions(h)
>>> f = L(lambda n: h[n]) - Integer(1)
>>> f(f.revert())
h[1] + O^8

ALGORITHM:

Let \(F\) be a symmetric function with valuation \(1\), i.e., whose constant term vanishes and whose degree one term equals \(b p_1\). Then

\[(F - b p_1) \circ G = F \circ G - b p_1 \circ G = p_1 - b G,\]

and therefore \(G = (p_1 - (F - b p_1) \circ G) / b\), which allows recursive computation of \(G\).

See also

The compositional inverse \(\Omega\) of the symmetric function \(h_1 + h_2 + \dots\) can be handled much more efficiently using specialized methods. See LogarithmCycleIndexSeries()

AUTHORS:

  • Andrew Gainer-Dewar

  • Martin Rubey

suspension()[source]

Return the suspension of self.

This is an involution, that maps the homogeneous component \(f_n\) of degree \(n\) to \((-1)^{n - 1} \omega(f_n)\), where \(omega\) is the usual involution of symmetric functions.

EXAMPLES:

sage: s = SymmetricFunctions(QQ).s()
sage: ls = LazySymmetricFunctions(s)
sage: f = ls(lambda n: s([n]), valuation=1)
sage: g = f.revert().suspension(); g
s[1] + (s[1,1]) + (s[2,1]) + (s[2,1,1]+s[3,1]) + ...
sage: g.revert().suspension()
s[1] + s[2] + s[3] + s[4] + s[5] + ...
>>> from sage.all import *
>>> s = SymmetricFunctions(QQ).s()
>>> ls = LazySymmetricFunctions(s)
>>> f = ls(lambda n: s([n]), valuation=Integer(1))
>>> g = f.revert().suspension(); g
s[1] + (s[1,1]) + (s[2,1]) + (s[2,1,1]+s[3,1]) + ...
>>> g.revert().suspension()
s[1] + s[2] + s[3] + s[4] + s[5] + ...
symmetric_function(degree=None)[source]

Return self as a symmetric function if self is actually so.

INPUT:

  • degreeNone or an integer

OUTPUT:

If degree is not None, the terms of the series of degree greater than degree are first truncated. If degree is None and the series is not a polynomial polynomial, a ValueError is raised.

EXAMPLES:

sage: # needs sage.modules
sage: s = SymmetricFunctions(QQ).s()
sage: S = LazySymmetricFunctions(s)
sage: elt = S(s[2])
sage: elt.symmetric_function()
s[2]
>>> from sage.all import *
>>> # needs sage.modules
>>> s = SymmetricFunctions(QQ).s()
>>> S = LazySymmetricFunctions(s)
>>> elt = S(s[Integer(2)])
>>> elt.symmetric_function()
s[2]