Lazy Series Rings#

We provide lazy implementations for various \(\NN\)-graded rings.

`LazyLaurentSeriesRing`	The ring of lazy Laurent series.
`LazyPowerSeriesRing`	The ring of (possibly multivariate) lazy Taylor series.
`LazyCompletionGradedAlgebra`	The completion of a graded algebra consisting of formal series.
`LazySymmetricFunctions`	The ring of (possibly multivariate) lazy symmetric functions.
`LazyDirichletSeriesRing`	The ring of lazy Dirichlet series.

Warning

When the halting precision is infinite, the default for bool(f) is True for any lazy series f that is not known to be zero. This could end up resulting in infinite loops:

Sage

sage: L.<x> = LazyPowerSeriesRing(ZZ)
sage: f = L(lambda n: 0, valuation=0)
sage: 1 / f  # not tested - infinite loop

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('x',)); (x,) = L._first_ngens(1)
>>> f = L(lambda n: Integer(0), valuation=Integer(0))
>>> Integer(1) / f  # not tested - infinite loop

See also

The examples of LazyLaurentSeriesRing contain a discussion about the different methods of comparisons the lazy series can use.

AUTHORS:

Kwankyu Lee (2019-02-24): initial version
Tejasvi Chebrolu, Martin Rubey, Travis Scrimshaw (2021-08): refactored and expanded functionality

class sage.rings.lazy_series_ring.LazyCompletionGradedAlgebra(basis, sparse=True, category=None)[source]#

Bases: LazySeriesRing

The completion of a graded algebra consisting of formal series.

For a graded algebra \(A\), we can form a completion of \(A\) consisting of all formal series of \(A\) such that each homogeneous component is a finite linear combination of basis elements of \(A\).

INPUT:

basis – a graded algebra
names – name(s) of the alphabets
sparse – (default: True) whether we use a sparse or a dense representation

EXAMPLES:

Sage

sage: # needs sage.modules
sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: S = NCSF.Complete()
sage: L = S.formal_series_ring(); L
Lazy completion of Non-Commutative Symmetric Functions
 over the Rational Field in the Complete basis
sage: f = 1 / (1 - L(S[1])); f
S[] + S[1] + (S[1,1]) + (S[1,1,1]) + (S[1,1,1,1]) + (S[1,1,1,1,1])
 + (S[1,1,1,1,1,1]) + O^7
sage: g = 1 / (1 - L(S[2])); g
S[] + S[2] + (S[2,2]) + (S[2,2,2]) + O^7
sage: f * g
S[] + S[1] + (S[1,1]+S[2]) + (S[1,1,1]+S[1,2])
 + (S[1,1,1,1]+S[1,1,2]+S[2,2]) + (S[1,1,1,1,1]+S[1,1,1,2]+S[1,2,2])
 + (S[1,1,1,1,1,1]+S[1,1,1,1,2]+S[1,1,2,2]+S[2,2,2]) + O^7
sage: g * f
S[] + S[1] + (S[1,1]+S[2]) + (S[1,1,1]+S[2,1])
 + (S[1,1,1,1]+S[2,1,1]+S[2,2]) + (S[1,1,1,1,1]+S[2,1,1,1]+S[2,2,1])
 + (S[1,1,1,1,1,1]+S[2,1,1,1,1]+S[2,2,1,1]+S[2,2,2]) + O^7
sage: f * g - g * f
(S[1,2]-S[2,1]) + (S[1,1,2]-S[2,1,1])
 + (S[1,1,1,2]+S[1,2,2]-S[2,1,1,1]-S[2,2,1])
 + (S[1,1,1,1,2]+S[1,1,2,2]-S[2,1,1,1,1]-S[2,2,1,1]) + O^7

Python

>>> from sage.all import *
>>> # needs sage.modules
>>> NCSF = NonCommutativeSymmetricFunctions(QQ)
>>> S = NCSF.Complete()
>>> L = S.formal_series_ring(); L
Lazy completion of Non-Commutative Symmetric Functions
 over the Rational Field in the Complete basis
>>> f = Integer(1) / (Integer(1) - L(S[Integer(1)])); f
S[] + S[1] + (S[1,1]) + (S[1,1,1]) + (S[1,1,1,1]) + (S[1,1,1,1,1])
 + (S[1,1,1,1,1,1]) + O^7
>>> g = Integer(1) / (Integer(1) - L(S[Integer(2)])); g
S[] + S[2] + (S[2,2]) + (S[2,2,2]) + O^7
>>> f * g
S[] + S[1] + (S[1,1]+S[2]) + (S[1,1,1]+S[1,2])
 + (S[1,1,1,1]+S[1,1,2]+S[2,2]) + (S[1,1,1,1,1]+S[1,1,1,2]+S[1,2,2])
 + (S[1,1,1,1,1,1]+S[1,1,1,1,2]+S[1,1,2,2]+S[2,2,2]) + O^7
>>> g * f
S[] + S[1] + (S[1,1]+S[2]) + (S[1,1,1]+S[2,1])
 + (S[1,1,1,1]+S[2,1,1]+S[2,2]) + (S[1,1,1,1,1]+S[2,1,1,1]+S[2,2,1])
 + (S[1,1,1,1,1,1]+S[2,1,1,1,1]+S[2,2,1,1]+S[2,2,2]) + O^7
>>> f * g - g * f
(S[1,2]-S[2,1]) + (S[1,1,2]-S[2,1,1])
 + (S[1,1,1,2]+S[1,2,2]-S[2,1,1,1]-S[2,2,1])
 + (S[1,1,1,1,2]+S[1,1,2,2]-S[2,1,1,1,1]-S[2,2,1,1]) + O^7

Element[source]#: alias of LazyCompletionGradedAlgebraElement

some_elements()[source]#

Return a list of elements of self.

EXAMPLES:

Sage

sage: m = SymmetricFunctions(GF(5)).m()                                     # needs sage.modules
sage: L = LazySymmetricFunctions(m)                                         # needs sage.modules
sage: L.some_elements()[:5]                                                 # needs sage.modules
[0, m[], 2*m[] + 2*m[1] + 3*m[2], 2*m[1] + 3*m[2],
 3*m[] + 2*m[1] + (m[1,1]+m[2])
       + (2*m[1,1,1]+m[3])
       + (2*m[1,1,1,1]+4*m[2,1,1]+2*m[2,2])
       + (3*m[2,1,1,1]+3*m[3,1,1]+4*m[3,2]+m[5])
       + (2*m[2,2,1,1]+m[2,2,2]+2*m[3,2,1]+2*m[3,3]+m[4,1,1]+3*m[4,2]+4*m[5,1]+4*m[6])
       + O^7]

sage: # needs sage.modules
sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: S = NCSF.Complete()
sage: L = S.formal_series_ring()
sage: L.some_elements()[:4]
[0, S[], 2*S[] + 2*S[1] + (3*S[1,1]), 2*S[1] + (3*S[1,1])]

Python

>>> from sage.all import *
>>> m = SymmetricFunctions(GF(Integer(5))).m()                                     # needs sage.modules
>>> L = LazySymmetricFunctions(m)                                         # needs sage.modules
>>> L.some_elements()[:Integer(5)]                                                 # needs sage.modules
[0, m[], 2*m[] + 2*m[1] + 3*m[2], 2*m[1] + 3*m[2],
 3*m[] + 2*m[1] + (m[1,1]+m[2])
       + (2*m[1,1,1]+m[3])
       + (2*m[1,1,1,1]+4*m[2,1,1]+2*m[2,2])
       + (3*m[2,1,1,1]+3*m[3,1,1]+4*m[3,2]+m[5])
       + (2*m[2,2,1,1]+m[2,2,2]+2*m[3,2,1]+2*m[3,3]+m[4,1,1]+3*m[4,2]+4*m[5,1]+4*m[6])
       + O^7]

>>> # needs sage.modules
>>> NCSF = NonCommutativeSymmetricFunctions(QQ)
>>> S = NCSF.Complete()
>>> L = S.formal_series_ring()
>>> L.some_elements()[:Integer(4)]
[0, S[], 2*S[] + 2*S[1] + (3*S[1,1]), 2*S[1] + (3*S[1,1])]

class sage.rings.lazy_series_ring.LazyDirichletSeriesRing(base_ring, names, sparse=True, category=None)[source]#

Bases: LazySeriesRing

The ring of lazy Dirichlet series.

INPUT:

base_ring – base ring of this Dirichlet series ring
names – name of the generator of this Dirichlet series ring
sparse – (default: True) whether this series is sparse or not

Unlike formal univariate Laurent/power series (over a field), the ring of formal Dirichlet series is not a Wikipedia article discrete_valuation_ring. On the other hand, it is a Wikipedia article local_ring. The unique maximal ideal consists of all non-invertible series, i.e., series with vanishing constant term.

Todo

According to the answers in https://mathoverflow.net/questions/5522/dirichlet-series-with-integer-coefficients-as-a-ufd, (which, in particular, references arXiv math/0105219) the ring of formal Dirichlet series is actually a Wikipedia article Unique_factorization_domain over \(\ZZ\).

Note

An interesting valuation is described in Emil Daniel Schwab; Gheorghe Silberberg A note on some discrete valuation rings of arithmetical functions, Archivum Mathematicum, Vol. 36 (2000), No. 2, 103-109, http://dml.cz/dmlcz/107723. Let \(J_k\) be the ideal of Dirichlet series whose coefficient \(f[n]\) of \(n^s\) vanishes if \(n\) has less than \(k\) prime factors, counting multiplicities. For any Dirichlet series \(f\), let \(D(f)\) be the largest integer \(k\) such that \(f\) is in \(J_k\). Then \(D\) is surjective, \(D(f g) = D(f) + D(g)\) for nonzero \(f\) and \(g\), and \(D(f + g) \geq \min(D(f), D(g))\) provided that \(f + g\) is nonzero.

For example, \(J_1\) are series with no constant term, and \(J_2\) are series such that \(f[1]\) and \(f[p]\) for prime \(p\) vanish.

Since this is a chain of increasing ideals, the ring of formal Dirichlet series is not a Wikipedia article Noetherian_ring.

Evidently, this valuation cannot be computed for a given series.

EXAMPLES:

Sage

sage: LazyDirichletSeriesRing(ZZ, 't')
Lazy Dirichlet Series Ring in t over Integer Ring

Python

>>> from sage.all import *
>>> LazyDirichletSeriesRing(ZZ, 't')
Lazy Dirichlet Series Ring in t over Integer Ring

The ideal generated by \(2^-s\) and \(3^-s\) is not principal:

Sage

sage: L = LazyDirichletSeriesRing(QQ, 's')
sage: L in PrincipalIdealDomains
False

Python

>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(QQ, 's')
>>> L in PrincipalIdealDomains
False

Element[source]#: alias of LazyDirichletSeries

one()[source]#

Return the constant series \(1\).

EXAMPLES:

Sage

sage: L = LazyDirichletSeriesRing(ZZ, 'z')
sage: L.one()                                                               # needs sage.symbolic
1
sage: ~L.one()                                                              # needs sage.symbolic
1 + O(1/(8^z))

Python

>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, 'z')
>>> L.one()                                                               # needs sage.symbolic
1
>>> ~L.one()                                                              # needs sage.symbolic
1 + O(1/(8^z))

some_elements()[source]#

Return a list of elements of self.

EXAMPLES:

Sage

sage: L = LazyDirichletSeriesRing(ZZ, 'z')
sage: l = L.some_elements()
sage: l                                                                     # needs sage.symbolic
[0, 1,
 1/(4^z) + 1/(5^z) + 1/(6^z) + O(1/(7^z)),
 1/(2^z) - 1/(3^z) + 2/4^z - 2/5^z + 3/6^z - 3/7^z + 4/8^z - 4/9^z,
 1/(2^z) - 1/(3^z) + 2/4^z - 2/5^z + 3/6^z - 3/7^z + 4/8^z - 4/9^z + 1/(10^z) + 1/(11^z) + 1/(12^z) + O(1/(13^z)),
 1 + 4/2^z + 9/3^z + 16/4^z + 25/5^z + 36/6^z + 49/7^z + O(1/(8^z))]

sage: L = LazyDirichletSeriesRing(QQ, 'z')
sage: l = L.some_elements()
sage: l                                                                     # needs sage.symbolic
[0, 1,
 1/2/4^z + 1/2/5^z + 1/2/6^z + O(1/(7^z)),
 1/2 - 1/2/2^z + 2/3^z - 2/4^z + 1/(6^z) - 1/(7^z) + 42/8^z + 2/3/9^z,
 1/2 - 1/2/2^z + 2/3^z - 2/4^z + 1/(6^z) - 1/(7^z) + 42/8^z + 2/3/9^z + 1/2/10^z + 1/2/11^z + 1/2/12^z + O(1/(13^z)),
 1 + 4/2^z + 9/3^z + 16/4^z + 25/5^z + 36/6^z + 49/7^z + O(1/(8^z))]

Python

>>> from sage.all import *
>>> L = LazyDirichletSeriesRing(ZZ, 'z')
>>> l = L.some_elements()
>>> l                                                                     # needs sage.symbolic
[0, 1,
 1/(4^z) + 1/(5^z) + 1/(6^z) + O(1/(7^z)),
 1/(2^z) - 1/(3^z) + 2/4^z - 2/5^z + 3/6^z - 3/7^z + 4/8^z - 4/9^z,
 1/(2^z) - 1/(3^z) + 2/4^z - 2/5^z + 3/6^z - 3/7^z + 4/8^z - 4/9^z + 1/(10^z) + 1/(11^z) + 1/(12^z) + O(1/(13^z)),
 1 + 4/2^z + 9/3^z + 16/4^z + 25/5^z + 36/6^z + 49/7^z + O(1/(8^z))]

>>> L = LazyDirichletSeriesRing(QQ, 'z')
>>> l = L.some_elements()
>>> l                                                                     # needs sage.symbolic
[0, 1,
 1/2/4^z + 1/2/5^z + 1/2/6^z + O(1/(7^z)),
 1/2 - 1/2/2^z + 2/3^z - 2/4^z + 1/(6^z) - 1/(7^z) + 42/8^z + 2/3/9^z,
 1/2 - 1/2/2^z + 2/3^z - 2/4^z + 1/(6^z) - 1/(7^z) + 42/8^z + 2/3/9^z + 1/2/10^z + 1/2/11^z + 1/2/12^z + O(1/(13^z)),
 1 + 4/2^z + 9/3^z + 16/4^z + 25/5^z + 36/6^z + 49/7^z + O(1/(8^z))]

class sage.rings.lazy_series_ring.LazyLaurentSeriesRing(base_ring, names, sparse=True, category=None)[source]#

Bases: LazySeriesRing

The ring of lazy Laurent series.

The ring of Laurent series over a ring with the usual arithmetic where the coefficients are computed lazily.

INPUT:

base_ring – base ring
names – name of the generator
sparse – (default: True) whether the implementation of the series is sparse or not

EXAMPLES:

Sage

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: 1 / (1 - z)
1 + z + z^2 + O(z^3)
sage: 1 / (1 - z) == 1 / (1 - z)
True
sage: L in Fields
True

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> Integer(1) / (Integer(1) - z)
1 + z + z^2 + O(z^3)
>>> Integer(1) / (Integer(1) - z) == Integer(1) / (Integer(1) - z)
True
>>> L in Fields
True

Lazy Laurent series ring over a finite field:

Sage

sage: # needs sage.rings.finite_rings
sage: L.<z> = LazyLaurentSeriesRing(GF(3)); L
Lazy Laurent Series Ring in z over Finite Field of size 3
sage: e = 1 / (1 + z)
sage: e.coefficient(100)
1
sage: e.coefficient(100).parent()
Finite Field of size 3

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> L = LazyLaurentSeriesRing(GF(Integer(3)), names=('z',)); (z,) = L._first_ngens(1); L
Lazy Laurent Series Ring in z over Finite Field of size 3
>>> e = Integer(1) / (Integer(1) + z)
>>> e.coefficient(Integer(100))
1
>>> e.coefficient(Integer(100)).parent()
Finite Field of size 3

Series can be defined by specifying a coefficient function and a valuation:

Sage

sage: R.<x,y> = QQ[]
sage: L.<z> = LazyLaurentSeriesRing(R)
sage: def coeff(n):
....:     if n < 0:
....:         return -2 + n
....:     if n == 0:
....:         return 6
....:     return x + y^n
sage: f = L(coeff, valuation=-5)
sage: f
-7*z^-5 - 6*z^-4 - 5*z^-3 - 4*z^-2 - 3*z^-1 + 6 + (x + y)*z + O(z^2)
sage: 1 / (1 - f)
1/7*z^5 - 6/49*z^6 + 1/343*z^7 + 8/2401*z^8 + 64/16807*z^9
 + 17319/117649*z^10 + (1/49*x + 1/49*y - 180781/823543)*z^11 + O(z^12)
sage: L(coeff, valuation=-3, degree=3, constant=x)
-5*z^-3 - 4*z^-2 - 3*z^-1 + 6 + (x + y)*z + (y^2 + x)*z^2
 + x*z^3 + x*z^4 + x*z^5 + O(z^6)

Python

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> L = LazyLaurentSeriesRing(R, names=('z',)); (z,) = L._first_ngens(1)
>>> def coeff(n):
...     if n < Integer(0):
...         return -Integer(2) + n
...     if n == Integer(0):
...         return Integer(6)
...     return x + y**n
>>> f = L(coeff, valuation=-Integer(5))
>>> f
-7*z^-5 - 6*z^-4 - 5*z^-3 - 4*z^-2 - 3*z^-1 + 6 + (x + y)*z + O(z^2)
>>> Integer(1) / (Integer(1) - f)
1/7*z^5 - 6/49*z^6 + 1/343*z^7 + 8/2401*z^8 + 64/16807*z^9
 + 17319/117649*z^10 + (1/49*x + 1/49*y - 180781/823543)*z^11 + O(z^12)
>>> L(coeff, valuation=-Integer(3), degree=Integer(3), constant=x)
-5*z^-3 - 4*z^-2 - 3*z^-1 + 6 + (x + y)*z + (y^2 + x)*z^2
 + x*z^3 + x*z^4 + x*z^5 + O(z^6)

We can also specify a polynomial or the initial coefficients. Additionally, we may specify that all coefficients are equal to a given constant, beginning at a given degree:

Sage

sage: L([1, x, y, 0, x+y])
1 + x*z + y*z^2 + (x + y)*z^4
sage: L([1, x, y, 0, x+y], constant=2)
1 + x*z + y*z^2 + (x + y)*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + O(z^8)
sage: L([1, x, y, 0, x+y], degree=7, constant=2)
1 + x*z + y*z^2 + (x + y)*z^4 + 2*z^7 + 2*z^8 + 2*z^9 + O(z^10)
sage: L([1, x, y, 0, x+y], valuation=-2)
z^-2 + x*z^-1 + y + (x + y)*z^2
sage: L([1, x, y, 0, x+y], valuation=-2, constant=3)
z^-2 + x*z^-1 + y + (x + y)*z^2 + 3*z^3 + 3*z^4 + 3*z^5 + O(z^6)
sage: L([1, x, y, 0, x+y], valuation=-2, degree=4, constant=3)
z^-2 + x*z^-1 + y + (x + y)*z^2 + 3*z^4 + 3*z^5 + 3*z^6 + O(z^7)

Python

>>> from sage.all import *
>>> L([Integer(1), x, y, Integer(0), x+y])
1 + x*z + y*z^2 + (x + y)*z^4
>>> L([Integer(1), x, y, Integer(0), x+y], constant=Integer(2))
1 + x*z + y*z^2 + (x + y)*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + O(z^8)
>>> L([Integer(1), x, y, Integer(0), x+y], degree=Integer(7), constant=Integer(2))
1 + x*z + y*z^2 + (x + y)*z^4 + 2*z^7 + 2*z^8 + 2*z^9 + O(z^10)
>>> L([Integer(1), x, y, Integer(0), x+y], valuation=-Integer(2))
z^-2 + x*z^-1 + y + (x + y)*z^2
>>> L([Integer(1), x, y, Integer(0), x+y], valuation=-Integer(2), constant=Integer(3))
z^-2 + x*z^-1 + y + (x + y)*z^2 + 3*z^3 + 3*z^4 + 3*z^5 + O(z^6)
>>> L([Integer(1), x, y, Integer(0), x+y], valuation=-Integer(2), degree=Integer(4), constant=Integer(3))
z^-2 + x*z^-1 + y + (x + y)*z^2 + 3*z^4 + 3*z^5 + 3*z^6 + O(z^7)

Some additional examples over the integer ring:

Sage

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: L in Fields
False
sage: 1 / (1 - 2*z)^3
1 + 6*z + 24*z^2 + 80*z^3 + 240*z^4 + 672*z^5 + 1792*z^6 + O(z^7)

sage: R.<x> = LaurentPolynomialRing(ZZ)
sage: L(x^-2 + 3 + x)
z^-2 + 3 + z
sage: L(x^-2 + 3 + x, valuation=-5, constant=2)
z^-5 + 3*z^-3 + z^-2 + 2*z^-1 + 2 + 2*z + O(z^2)
sage: L(x^-2 + 3 + x, valuation=-5, degree=0, constant=2)
z^-5 + 3*z^-3 + z^-2 + 2 + 2*z + 2*z^2 + O(z^3)

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> L in Fields
False
>>> Integer(1) / (Integer(1) - Integer(2)*z)**Integer(3)
1 + 6*z + 24*z^2 + 80*z^3 + 240*z^4 + 672*z^5 + 1792*z^6 + O(z^7)

>>> R = LaurentPolynomialRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> L(x**-Integer(2) + Integer(3) + x)
z^-2 + 3 + z
>>> L(x**-Integer(2) + Integer(3) + x, valuation=-Integer(5), constant=Integer(2))
z^-5 + 3*z^-3 + z^-2 + 2*z^-1 + 2 + 2*z + O(z^2)
>>> L(x**-Integer(2) + Integer(3) + x, valuation=-Integer(5), degree=Integer(0), constant=Integer(2))
z^-5 + 3*z^-3 + z^-2 + 2 + 2*z + 2*z^2 + O(z^3)

We can truncate a series, shift its coefficients, or replace all coefficients beginning with a given degree by a constant:

Sage

sage: f = 1 / (z + z^2)
sage: f
z^-1 - 1 + z - z^2 + z^3 - z^4 + z^5 + O(z^6)
sage: L(f, valuation=2)
z^2 - z^3 + z^4 - z^5 + z^6 - z^7 + z^8 + O(z^9)
sage: L(f, degree=3)
z^-1 - 1 + z - z^2
sage: L(f, degree=3, constant=2)
z^-1 - 1 + z - z^2 + 2*z^3 + 2*z^4 + 2*z^5 + O(z^6)
sage: L(f, valuation=1, degree=4)
z - z^2 + z^3
sage: L(f, valuation=1, degree=4, constant=5)
z - z^2 + z^3 + 5*z^4 + 5*z^5 + 5*z^6 + O(z^7)

Python

>>> from sage.all import *
>>> f = Integer(1) / (z + z**Integer(2))
>>> f
z^-1 - 1 + z - z^2 + z^3 - z^4 + z^5 + O(z^6)
>>> L(f, valuation=Integer(2))
z^2 - z^3 + z^4 - z^5 + z^6 - z^7 + z^8 + O(z^9)
>>> L(f, degree=Integer(3))
z^-1 - 1 + z - z^2
>>> L(f, degree=Integer(3), constant=Integer(2))
z^-1 - 1 + z - z^2 + 2*z^3 + 2*z^4 + 2*z^5 + O(z^6)
>>> L(f, valuation=Integer(1), degree=Integer(4))
z - z^2 + z^3
>>> L(f, valuation=Integer(1), degree=Integer(4), constant=Integer(5))
z - z^2 + z^3 + 5*z^4 + 5*z^5 + 5*z^6 + O(z^7)

Power series can be defined recursively (see sage.rings.lazy_series.LazyModuleElement.define() for more examples):

Sage

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: s = L.undefined(valuation=0)
sage: s.define(1 + z*s^2)
sage: s
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> s = L.undefined(valuation=Integer(0))
>>> s.define(Integer(1) + z*s**Integer(2))
>>> s
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)

By default, any two series f and g that are not known to be equal are considered to be different:

Sage

sage: f = L(lambda n: 0, valuation=0)
sage: f == 0
False

sage: f = L(constant=1, valuation=0).derivative(); f
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)
sage: g = L(lambda n: (n+1), valuation=0); g
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)
sage: f == g
False

Python

>>> from sage.all import *
>>> f = L(lambda n: Integer(0), valuation=Integer(0))
>>> f == Integer(0)
False

>>> f = L(constant=Integer(1), valuation=Integer(0)).derivative(); f
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)
>>> g = L(lambda n: (n+Integer(1)), valuation=Integer(0)); g
1 + 2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)
>>> f == g
False

Warning

We have imposed that (f == g) == not (f != g), and so f != g returning True might not mean that the two series are actually different:

Sage

sage: f = L(lambda n: 0, valuation=0)
sage: g = L.zero()
sage: f != g
True

Python

>>> from sage.all import *
>>> f = L(lambda n: Integer(0), valuation=Integer(0))
>>> g = L.zero()
>>> f != g
True

This can be verified by is_nonzero(), which only returns True if the series is known to be nonzero:

Sage

sage: (f - g).is_nonzero()
False

Python

>>> from sage.all import *
>>> (f - g).is_nonzero()
False

The implementation of the ring can be either be a sparse or a dense one. The default is a sparse implementation:

Sage

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: L.is_sparse()
True
sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: L.is_sparse()
False

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> L.is_sparse()
True
>>> L = LazyLaurentSeriesRing(ZZ, sparse=False, names=('z',)); (z,) = L._first_ngens(1)
>>> L.is_sparse()
False

We additionally provide two other methods of performing comparisons. The first is raising a ValueError and the second uses a check up to a (user set) finite precision. These behaviors are set using the options secure and halting_precision. In particular, this applies to series that are not specified by a finite number of initial coefficients and a constant for the remaining coefficients. Equality checking will depend on the coefficients which have already been computed. If this information is not enough to check that two series are different, then if L.options.secure is set to True, then we raise a ValueError:

Sage

sage: L.options.secure = True
sage: f = 1 / (z + z^2); f
z^-1 - 1 + z - z^2 + z^3 - z^4 + z^5 + O(z^6)
sage: f2 = f * 2  # currently no coefficients computed
sage: f3 = f * 3  # currently no coefficients computed
sage: f2 == f3
Traceback (most recent call last):
...
ValueError: undecidable
sage: f2  # computes some of the coefficients of f2
2*z^-1 - 2 + 2*z - 2*z^2 + 2*z^3 - 2*z^4 + 2*z^5 + O(z^6)
sage: f3  # computes some of the coefficients of f3
3*z^-1 - 3 + 3*z - 3*z^2 + 3*z^3 - 3*z^4 + 3*z^5 + O(z^6)
sage: f2 == f3
False
sage: f2a = f + f
sage: f2 == f2a
Traceback (most recent call last):
...
ValueError: undecidable
sage: zf = L(lambda n: 0, valuation=0)
sage: zf == 0
Traceback (most recent call last):
...
ValueError: undecidable

Python

>>> from sage.all import *
>>> L.options.secure = True
>>> f = Integer(1) / (z + z**Integer(2)); f
z^-1 - 1 + z - z^2 + z^3 - z^4 + z^5 + O(z^6)
>>> f2 = f * Integer(2)  # currently no coefficients computed
>>> f3 = f * Integer(3)  # currently no coefficients computed
>>> f2 == f3
Traceback (most recent call last):
...
ValueError: undecidable
>>> f2  # computes some of the coefficients of f2
2*z^-1 - 2 + 2*z - 2*z^2 + 2*z^3 - 2*z^4 + 2*z^5 + O(z^6)
>>> f3  # computes some of the coefficients of f3
3*z^-1 - 3 + 3*z - 3*z^2 + 3*z^3 - 3*z^4 + 3*z^5 + O(z^6)
>>> f2 == f3
False
>>> f2a = f + f
>>> f2 == f2a
Traceback (most recent call last):
...
ValueError: undecidable
>>> zf = L(lambda n: Integer(0), valuation=Integer(0))
>>> zf == Integer(0)
Traceback (most recent call last):
...
ValueError: undecidable

For boolean checks, an error is raised when it is not known to be nonzero:

Sage

sage: bool(zf)
Traceback (most recent call last):
...
ValueError: undecidable

Python

>>> from sage.all import *
>>> bool(zf)
Traceback (most recent call last):
...
ValueError: undecidable

If the halting precision is set to a finite number \(p\) (for unlimited precision, it is set to None), then it will check up to \(p\) values from the current position:

Sage

sage: L.options.halting_precision = 20
sage: f2 = f * 2  # currently no coefficients computed
sage: f3 = f * 3  # currently no coefficients computed
sage: f2 == f3
False
sage: f2a = f + f
sage: f2 == f2a
True
sage: zf = L(lambda n: 0, valuation=0)
sage: zf == 0
True

Python

>>> from sage.all import *
>>> L.options.halting_precision = Integer(20)
>>> f2 = f * Integer(2)  # currently no coefficients computed
>>> f3 = f * Integer(3)  # currently no coefficients computed
>>> f2 == f3
False
>>> f2a = f + f
>>> f2 == f2a
True
>>> zf = L(lambda n: Integer(0), valuation=Integer(0))
>>> zf == Integer(0)
True

Element[source]#: alias of LazyLaurentSeries

euler()[source]#

Return the Euler function as an element of 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

sage: L.<q> = LazyLaurentSeriesRing(ZZ)
sage: phi = q.euler()
sage: phi
1 - q - q^2 + q^5 + O(q^7)

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('q',)); (q,) = L._first_ngens(1)
>>> phi = q.euler()
>>> phi
1 - q - q^2 + q^5 + O(q^7)

We verify that \(1 / phi\) gives the generating function for all partitions:

Sage

sage: P = 1 / phi; P
1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + O(q^7)
sage: P[:20] == [Partitions(n).cardinality() for n in range(20)]            # needs sage.libs.flint
True

Python

>>> from sage.all import *
>>> P = Integer(1) / phi; P
1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + O(q^7)
>>> P[:Integer(20)] == [Partitions(n).cardinality() for n in range(Integer(20))]            # needs sage.libs.flint
True

REFERENCES:

Wikipedia article Euler_function

gen(n=0)[source]#

Return the n-th generator of self.

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.gen()
z
sage: L.gen(3)
Traceback (most recent call last):
...
IndexError: there is only one generator

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> L.gen()
z
>>> L.gen(Integer(3))
Traceback (most recent call last):
...
IndexError: there is only one generator

gens()[source]#

Return the generators of self.

EXAMPLES:

Sage

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

Python

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

ngens()[source]#

Return the number of generators of self.

This is always 1.

EXAMPLES:

Sage

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: L.ngens()
1

Python

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

q_pochhammer(q=None)[source]#

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

sage: q = ZZ['q'].fraction_field().gen()
sage: L.<z> = LazyLaurentSeriesRing(q.parent())
sage: qpoch = L.q_pochhammer(q)
sage: qpoch
1
 + (-1/(-q + 1))*z
 + (q/(q^3 - q^2 - q + 1))*z^2
 + (-q^3/(-q^6 + q^5 + q^4 - q^2 - q + 1))*z^3
 + (q^6/(q^10 - q^9 - q^8 + 2*q^5 - q^2 - q + 1))*z^4
 + (-q^10/(-q^15 + q^14 + q^13 - q^10 - q^9 - q^8 + q^7 + q^6 + q^5 - q^2 - q + 1))*z^5
 + (q^15/(q^21 - q^20 - q^19 + q^16 + 2*q^14 - q^12 - q^11 - q^10 - q^9 + 2*q^7 + q^5 - q^2 - q + 1))*z^6
 + O(z^7)

Python

>>> from sage.all import *
>>> q = ZZ['q'].fraction_field().gen()
>>> L = LazyLaurentSeriesRing(q.parent(), names=('z',)); (z,) = L._first_ngens(1)
>>> qpoch = L.q_pochhammer(q)
>>> qpoch
1
 + (-1/(-q + 1))*z
 + (q/(q^3 - q^2 - q + 1))*z^2
 + (-q^3/(-q^6 + q^5 + q^4 - q^2 - q + 1))*z^3
 + (q^6/(q^10 - q^9 - q^8 + 2*q^5 - q^2 - q + 1))*z^4
 + (-q^10/(-q^15 + q^14 + q^13 - q^10 - q^9 - q^8 + q^7 + q^6 + q^5 - q^2 - q + 1))*z^5
 + (q^15/(q^21 - q^20 - q^19 + q^16 + 2*q^14 - q^12 - q^11 - q^10 - q^9 + 2*q^7 + q^5 - q^2 - q + 1))*z^6
 + O(z^7)

We show that \((z; q)_n = \frac{(z; q)_{\infty}}{(q^n z; q)_{\infty}}\):

Sage

sage: qpoch / qpoch(q*z)
1 - z + O(z^7)
sage: qpoch / qpoch(q^2*z)
1 + (-q - 1)*z + q*z^2 + O(z^7)
sage: qpoch / qpoch(q^3*z)
1 + (-q^2 - q - 1)*z + (q^3 + q^2 + q)*z^2 - q^3*z^3 + O(z^7)
sage: qpoch / qpoch(q^4*z)
1 + (-q^3 - q^2 - q - 1)*z + (q^5 + q^4 + 2*q^3 + q^2 + q)*z^2
 + (-q^6 - q^5 - q^4 - q^3)*z^3 + q^6*z^4 + O(z^7)

Python

>>> from sage.all import *
>>> qpoch / qpoch(q*z)
1 - z + O(z^7)
>>> qpoch / qpoch(q**Integer(2)*z)
1 + (-q - 1)*z + q*z^2 + O(z^7)
>>> qpoch / qpoch(q**Integer(3)*z)
1 + (-q^2 - q - 1)*z + (q^3 + q^2 + q)*z^2 - q^3*z^3 + O(z^7)
>>> qpoch / qpoch(q**Integer(4)*z)
1 + (-q^3 - q^2 - q - 1)*z + (q^5 + q^4 + 2*q^3 + q^2 + q)*z^2
 + (-q^6 - q^5 - q^4 - q^3)*z^3 + q^6*z^4 + O(z^7)

We can also construct part of Euler’s function:

Sage

sage: M.<a> = LazyLaurentSeriesRing(QQ)
sage: phi = sum(qpoch[i](q=a)*a^i for i in range(10))
sage: phi[:20] == M.euler()[:20]
True

Python

>>> from sage.all import *
>>> M = LazyLaurentSeriesRing(QQ, names=('a',)); (a,) = M._first_ngens(1)
>>> phi = sum(qpoch[i](q=a)*a**i for i in range(Integer(10)))
>>> phi[:Integer(20)] == M.euler()[:Integer(20)]
True

REFERENCES:

residue_field()[source]#

Return the residue field of the ring of integers of self.

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(QQ, 'z')
sage: L.residue_field()
Rational Field

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, 'z')
>>> L.residue_field()
Rational Field

series(coefficient, valuation, degree=None, constant=None)[source]#

Return a lazy Laurent series.

INPUT:

coefficient – Python function that computes coefficients or a list
valuation – integer; approximate valuation of the series
degree – (optional) integer
constant – (optional) an element of the base ring

Let the coefficient of index \(i\) mean the coefficient of the term of the series with exponent \(i\).

Python function coefficient returns the value of the coefficient of index \(i\) from input \(s\) and \(i\) where \(s\) is the series itself.

Let valuation be \(n\). All coefficients of index below \(n\) are zero. If constant is not specified, then the coefficient function is responsible to compute the values of all coefficients of index \(\ge n\). If degree or constant is a pair \((c,m)\), then the coefficient function is responsible to compute the values of all coefficients of index \(\ge n\) and \(< m\) and all the coefficients of index \(\ge m\) is the constant \(c\).

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.series(lambda s, i: i, 5, (1,10))
5*z^5 + 6*z^6 + 7*z^7 + 8*z^8 + 9*z^9 + z^10 + z^11 + z^12 + O(z^13)

sage: def g(s, i):
....:     if i < 0:
....:         return 1
....:     else:
....:         return s.coefficient(i - 1) + i
sage: e = L.series(g, -5); e
z^-5 + z^-4 + z^-3 + z^-2 + z^-1 + 1 + 2*z + O(z^2)
sage: f = e^-1; f
z^5 - z^6 - z^11 + O(z^12)
sage: f.coefficient(10)
0
sage: f.coefficient(20)
9
sage: f.coefficient(30)
-219

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> L.series(lambda s, i: i, Integer(5), (Integer(1),Integer(10)))
5*z^5 + 6*z^6 + 7*z^7 + 8*z^8 + 9*z^9 + z^10 + z^11 + z^12 + O(z^13)

>>> def g(s, i):
...     if i < Integer(0):
...         return Integer(1)
...     else:
...         return s.coefficient(i - Integer(1)) + i
>>> e = L.series(g, -Integer(5)); e
z^-5 + z^-4 + z^-3 + z^-2 + z^-1 + 1 + 2*z + O(z^2)
>>> f = e**-Integer(1); f
z^5 - z^6 - z^11 + O(z^12)
>>> f.coefficient(Integer(10))
0
>>> f.coefficient(Integer(20))
9
>>> f.coefficient(Integer(30))
-219

Alternatively, the coefficient can be a list of elements of the base ring. Then these elements are read as coefficients of the terms of degrees starting from the valuation. In this case, constant may be just an element of the base ring instead of a tuple or can be simply omitted if it is zero.

Sage

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: f = L.series([1,2,3,4], -5); f
z^-5 + 2*z^-4 + 3*z^-3 + 4*z^-2
sage: g = L.series([1,3,5,7,9], 5, constant=-1); g
z^5 + 3*z^6 + 5*z^7 + 7*z^8 + 9*z^9 - z^10 - z^11 - z^12 + O(z^13)

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> f = L.series([Integer(1),Integer(2),Integer(3),Integer(4)], -Integer(5)); f
z^-5 + 2*z^-4 + 3*z^-3 + 4*z^-2
>>> g = L.series([Integer(1),Integer(3),Integer(5),Integer(7),Integer(9)], Integer(5), constant=-Integer(1)); g
z^5 + 3*z^6 + 5*z^7 + 7*z^8 + 9*z^9 - z^10 - z^11 - z^12 + O(z^13)

some_elements()[source]#

Return a list of elements of self.

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.some_elements()[:7]
[0, 1, z,
 -3*z^-4 + z^-3 - 12*z^-2 - 2*z^-1 - 10 - 8*z + z^2 + z^3,
 z^-2 + z^3 + z^4 + z^5 + O(z^6),
 -2*z^-3 - 2*z^-2 + 4*z^-1 + 11 - z - 34*z^2 - 31*z^3 + O(z^4),
 4*z^-2 + z^-1 + z + 4*z^2 + 9*z^3 + 16*z^4 + O(z^5)]

sage: L = LazyLaurentSeriesRing(GF(2), 'z')
sage: L.some_elements()[:7]
[0, 1, z,
 z^-4 + z^-3 + z^2 + z^3,
 z^-2,
 1 + z + z^3 + z^4 + z^6 + O(z^7),
 z^-1 + z + z^3 + O(z^5)]

sage: L = LazyLaurentSeriesRing(GF(3), 'z')
sage: L.some_elements()[:7]
[0, 1, z,
 z^-3 + z^-1 + 2 + z + z^2 + z^3,
 z^-2,
 z^-3 + z^-2 + z^-1 + 2 + 2*z + 2*z^2 + O(z^3),
 z^-2 + z^-1 + z + z^2 + z^4 + O(z^5)]

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> L.some_elements()[:Integer(7)]
[0, 1, z,
 -3*z^-4 + z^-3 - 12*z^-2 - 2*z^-1 - 10 - 8*z + z^2 + z^3,
 z^-2 + z^3 + z^4 + z^5 + O(z^6),
 -2*z^-3 - 2*z^-2 + 4*z^-1 + 11 - z - 34*z^2 - 31*z^3 + O(z^4),
 4*z^-2 + z^-1 + z + 4*z^2 + 9*z^3 + 16*z^4 + O(z^5)]

>>> L = LazyLaurentSeriesRing(GF(Integer(2)), 'z')
>>> L.some_elements()[:Integer(7)]
[0, 1, z,
 z^-4 + z^-3 + z^2 + z^3,
 z^-2,
 1 + z + z^3 + z^4 + z^6 + O(z^7),
 z^-1 + z + z^3 + O(z^5)]

>>> L = LazyLaurentSeriesRing(GF(Integer(3)), 'z')
>>> L.some_elements()[:Integer(7)]
[0, 1, z,
 z^-3 + z^-1 + 2 + z + z^2 + z^3,
 z^-2,
 z^-3 + z^-2 + z^-1 + 2 + 2*z + 2*z^2 + O(z^3),
 z^-2 + z^-1 + z + z^2 + z^4 + O(z^5)]

taylor(f)[source]#

Return the Taylor expansion around \(0\) of the function f.

INPUT:

f – a function such that one of the following works:
- the substitution \(f(z)\), where \(z\) is a generator of self
- \(f\) is a function of a single variable with no poles at \(0\) and has a derivative method

EXAMPLES:

Sage

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: x = SR.var('x')
sage: f(x) = (1 + x) / (1 - x^2)
sage: L.taylor(f)
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> x = SR.var('x')
>>> __tmp__=var("x"); f = symbolic_expression((Integer(1) + x) / (Integer(1) - x**Integer(2))).function(x)
>>> L.taylor(f)
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)

For inputs as symbolic functions/expressions, the function must not have any poles at \(0\):

Sage

sage: f(x) = (1 + x^2) / sin(x^2)
sage: L.taylor(f)
<repr(...) failed: ValueError: power::eval(): division by zero>
sage: def g(a): return (1 + a^2) / sin(a^2)
sage: L.taylor(g)
z^-2 + 1 + 1/6*z^2 + 1/6*z^4 + O(z^5)

Python

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression((Integer(1) + x**Integer(2)) / sin(x**Integer(2))).function(x)
>>> L.taylor(f)
<repr(...) failed: ValueError: power::eval(): division by zero>
>>> def g(a): return (Integer(1) + a**Integer(2)) / sin(a**Integer(2))
>>> L.taylor(g)
z^-2 + 1 + 1/6*z^2 + 1/6*z^4 + O(z^5)

uniformizer()[source]#

Return a uniformizer of self..

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(QQ, 'z')
sage: L.uniformizer()
z

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, 'z')
>>> L.uniformizer()
z

class sage.rings.lazy_series_ring.LazyPowerSeriesRing(base_ring, names, sparse=True, category=None)[source]#

Bases: LazySeriesRing

The ring of (possibly multivariate) lazy Taylor series.

INPUT:

base_ring – base ring of this Taylor series ring
names – name(s) of the generator of this Taylor series ring
sparse – (default: True) whether this series is sparse or not

EXAMPLES:

Sage

sage: LazyPowerSeriesRing(ZZ, 't')
Lazy Taylor Series Ring in t over Integer Ring

sage: L.<x, y> = LazyPowerSeriesRing(QQ); L
Multivariate Lazy Taylor Series Ring in x, y over Rational Field

Python

>>> from sage.all import *
>>> LazyPowerSeriesRing(ZZ, 't')
Lazy Taylor Series Ring in t over Integer Ring

>>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2); L
Multivariate Lazy Taylor Series Ring in x, y over Rational Field

Element[source]#: alias of LazyPowerSeries

fraction_field()[source]#

Return the fraction field of self.

If this is with a single variable over a field, then the fraction field is the field of (lazy) formal Laurent series.

Todo

Implement other fraction fields.

EXAMPLES:

Sage

sage: L.<x> = LazyPowerSeriesRing(QQ)
sage: L.fraction_field()
Lazy Laurent Series Ring in x over Rational Field

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('x',)); (x,) = L._first_ngens(1)
>>> L.fraction_field()
Lazy Laurent Series Ring in x over Rational Field

gen(n=0)[source]#

Return the n-th generator of self.

EXAMPLES:

Sage

sage: L = LazyPowerSeriesRing(ZZ, 'z')
sage: L.gen()
z
sage: L.gen(3)
Traceback (most recent call last):
...
IndexError: there is only one generator

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, 'z')
>>> L.gen()
z
>>> L.gen(Integer(3))
Traceback (most recent call last):
...
IndexError: there is only one generator

gens()[source]#

Return the generators of self.

EXAMPLES:

Sage

sage: L = LazyPowerSeriesRing(ZZ, 'x,y')
sage: L.gens()
(x, y)

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, 'x,y')
>>> L.gens()
(x, y)

ngens()[source]#

Return the number of generators of self.

EXAMPLES:

Sage

sage: L.<z> = LazyPowerSeriesRing(ZZ)
sage: L.ngens()
1

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, names=('z',)); (z,) = L._first_ngens(1)
>>> L.ngens()
1

residue_field()[source]#

Return the residue field of the ring of integers of self.

EXAMPLES:

Sage

sage: L = LazyPowerSeriesRing(QQ, 'x')
sage: L.residue_field()
Rational Field

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, 'x')
>>> L.residue_field()
Rational Field

some_elements()[source]#

Return a list of elements of self.

EXAMPLES:

Sage

sage: L = LazyPowerSeriesRing(ZZ, 'z')
sage: L.some_elements()[:6]
[0, 1, z + z^2 + z^3 + O(z^4),
 -12 - 8*z + z^2 + z^3,
 1 + z - 2*z^2 - 7*z^3 - z^4 + 20*z^5 + 23*z^6 + O(z^7),
 z + 4*z^2 + 9*z^3 + 16*z^4 + 25*z^5 + 36*z^6 + O(z^7)]

sage: L = LazyPowerSeriesRing(GF(3)["q"], 'z')
sage: L.some_elements()[:6]
[0, 1, z + q*z^2 + q*z^3 + q*z^4 + O(z^5),
 z + z^2 + z^3,
 1 + z + z^2 + 2*z^3 + 2*z^4 + 2*z^5 + O(z^6),
 z + z^2 + z^4 + z^5 + O(z^7)]

sage: L = LazyPowerSeriesRing(GF(3), 'q, t')
sage: L.some_elements()[:6]
[0, 1, q,
 q + q^2 + q^3,
 1 + q + q^2 + (-q^3) + (-q^4) + (-q^5) + (-q^6) + O(q,t)^7,
 1 + (q+t) + (q^2-q*t+t^2) + (q^3+t^3)
   + (q^4+q^3*t+q*t^3+t^4)
   + (q^5-q^4*t+q^3*t^2+q^2*t^3-q*t^4+t^5)
   + (q^6-q^3*t^3+t^6) + O(q,t)^7]

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(ZZ, 'z')
>>> L.some_elements()[:Integer(6)]
[0, 1, z + z^2 + z^3 + O(z^4),
 -12 - 8*z + z^2 + z^3,
 1 + z - 2*z^2 - 7*z^3 - z^4 + 20*z^5 + 23*z^6 + O(z^7),
 z + 4*z^2 + 9*z^3 + 16*z^4 + 25*z^5 + 36*z^6 + O(z^7)]

>>> L = LazyPowerSeriesRing(GF(Integer(3))["q"], 'z')
>>> L.some_elements()[:Integer(6)]
[0, 1, z + q*z^2 + q*z^3 + q*z^4 + O(z^5),
 z + z^2 + z^3,
 1 + z + z^2 + 2*z^3 + 2*z^4 + 2*z^5 + O(z^6),
 z + z^2 + z^4 + z^5 + O(z^7)]

>>> L = LazyPowerSeriesRing(GF(Integer(3)), 'q, t')
>>> L.some_elements()[:Integer(6)]
[0, 1, q,
 q + q^2 + q^3,
 1 + q + q^2 + (-q^3) + (-q^4) + (-q^5) + (-q^6) + O(q,t)^7,
 1 + (q+t) + (q^2-q*t+t^2) + (q^3+t^3)
   + (q^4+q^3*t+q*t^3+t^4)
   + (q^5-q^4*t+q^3*t^2+q^2*t^3-q*t^4+t^5)
   + (q^6-q^3*t^3+t^6) + O(q,t)^7]

taylor(f)[source]#

Return the Taylor expansion around \(0\) of the function f.

INPUT:

f – a function such that one of the following works:
- the substitution \(f(z_1, \ldots, z_n)\), where \((z_1, \ldots, z_n)\) are the generators of self
- \(f\) is a function with no poles at \(0\) and has a derivative method

Warning

For inputs as symbolic functions/expressions, this does not check that the function does not have poles at \(0\).

EXAMPLES:

Sage

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: x = SR.var('x')
sage: f(x) = (1 + x) / (1 - x^3)
sage: L.taylor(f)
1 + z + z^3 + z^4 + z^6 + O(z^7)
sage: (1 + z) / (1 - z^3)
1 + z + z^3 + z^4 + z^6 + O(z^7)
sage: f(x) = cos(x + pi/2)
sage: L.taylor(f)
-z + 1/6*z^3 - 1/120*z^5 + O(z^7)

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> x = SR.var('x')
>>> __tmp__=var("x"); f = symbolic_expression((Integer(1) + x) / (Integer(1) - x**Integer(3))).function(x)
>>> L.taylor(f)
1 + z + z^3 + z^4 + z^6 + O(z^7)
>>> (Integer(1) + z) / (Integer(1) - z**Integer(3))
1 + z + z^3 + z^4 + z^6 + O(z^7)
>>> __tmp__=var("x"); f = symbolic_expression(cos(x + pi/Integer(2))).function(x)
>>> L.taylor(f)
-z + 1/6*z^3 - 1/120*z^5 + O(z^7)

For inputs as symbolic functions/expressions, the function must not have any poles at \(0\):

Sage

sage: L.<z> = LazyPowerSeriesRing(QQ, sparse=True)
sage: f = 1 / sin(x)
sage: L.taylor(f)
<repr(...) failed: ValueError: power::eval(): division by zero>

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, sparse=True, names=('z',)); (z,) = L._first_ngens(1)
>>> f = Integer(1) / sin(x)
>>> L.taylor(f)
<repr(...) failed: ValueError: power::eval(): division by zero>

Different multivariate inputs:

Sage

sage: L.<a,b> = LazyPowerSeriesRing(QQ)
sage: def f(x, y): return (1 + x) / (1 + y)
sage: L.taylor(f)
1 + (a-b) + (-a*b+b^2) + (a*b^2-b^3) + (-a*b^3+b^4) + (a*b^4-b^5) + (-a*b^5+b^6) + O(a,b)^7
sage: g(w, z) = (1 + w) / (1 + z)
sage: L.taylor(g)
1 + (a-b) + (-a*b+b^2) + (a*b^2-b^3) + (-a*b^3+b^4) + (a*b^4-b^5) + (-a*b^5+b^6) + O(a,b)^7
sage: y = SR.var('y')
sage: h = (1 + x) / (1 + y)
sage: L.taylor(h)
1 + (a-b) + (-a*b+b^2) + (a*b^2-b^3) + (-a*b^3+b^4) + (a*b^4-b^5) + (-a*b^5+b^6) + O(a,b)^7

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('a', 'b',)); (a, b,) = L._first_ngens(2)
>>> def f(x, y): return (Integer(1) + x) / (Integer(1) + y)
>>> L.taylor(f)
1 + (a-b) + (-a*b+b^2) + (a*b^2-b^3) + (-a*b^3+b^4) + (a*b^4-b^5) + (-a*b^5+b^6) + O(a,b)^7
>>> __tmp__=var("w,z"); g = symbolic_expression((Integer(1) + w) / (Integer(1) + z)).function(w,z)
>>> L.taylor(g)
1 + (a-b) + (-a*b+b^2) + (a*b^2-b^3) + (-a*b^3+b^4) + (a*b^4-b^5) + (-a*b^5+b^6) + O(a,b)^7
>>> y = SR.var('y')
>>> h = (Integer(1) + x) / (Integer(1) + y)
>>> L.taylor(h)
1 + (a-b) + (-a*b+b^2) + (a*b^2-b^3) + (-a*b^3+b^4) + (a*b^4-b^5) + (-a*b^5+b^6) + O(a,b)^7

uniformizer()[source]#

Return a uniformizer of self.

EXAMPLES:

Sage

sage: L = LazyPowerSeriesRing(QQ, 'x')
sage: L.uniformizer()
x

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, 'x')
>>> L.uniformizer()
x

class sage.rings.lazy_series_ring.LazySeriesRing[source]#

Bases: UniqueRepresentation, Parent

Abstract base class for lazy series.

characteristic()[source]#

Return the characteristic of this lazy power series ring, which is the same as the characteristic of its base ring.

EXAMPLES:

Sage

sage: L.<t> = LazyLaurentSeriesRing(ZZ)
sage: L.characteristic()
0

sage: R.<w> = LazyLaurentSeriesRing(GF(11)); R
Lazy Laurent Series Ring in w over Finite Field of size 11
sage: R.characteristic()
11

sage: R.<x, y> = LazyPowerSeriesRing(GF(7)); R
Multivariate Lazy Taylor Series Ring in x, y over Finite Field of size 7
sage: R.characteristic()
7

sage: L = LazyDirichletSeriesRing(ZZ, "s")
sage: L.characteristic()
0

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, names=('t',)); (t,) = L._first_ngens(1)
>>> L.characteristic()
0

>>> R = LazyLaurentSeriesRing(GF(Integer(11)), names=('w',)); (w,) = R._first_ngens(1); R
Lazy Laurent Series Ring in w over Finite Field of size 11
>>> R.characteristic()
11

>>> R = LazyPowerSeriesRing(GF(Integer(7)), names=('x', 'y',)); (x, y,) = R._first_ngens(2); R
Multivariate Lazy Taylor Series Ring in x, y over Finite Field of size 7
>>> R.characteristic()
7

>>> L = LazyDirichletSeriesRing(ZZ, "s")
>>> L.characteristic()
0

is_exact()[source]#

Return if self is exact or not.

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.is_exact()
True
sage: L = LazyLaurentSeriesRing(RR, 'z')
sage: L.is_exact()
False

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> L.is_exact()
True
>>> L = LazyLaurentSeriesRing(RR, 'z')
>>> L.is_exact()
False

is_sparse()[source]#

Return whether self is sparse or not.

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(ZZ, 'z', sparse=False)
sage: L.is_sparse()
False

sage: L = LazyLaurentSeriesRing(ZZ, 'z', sparse=True)
sage: L.is_sparse()
True

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z', sparse=False)
>>> L.is_sparse()
False

>>> L = LazyLaurentSeriesRing(ZZ, 'z', sparse=True)
>>> L.is_sparse()
True

one()[source]#

Return the constant series \(1\).

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.one()
1

sage: L = LazyPowerSeriesRing(ZZ, 'z')
sage: L.one()
1

sage: m = SymmetricFunctions(ZZ).m()                                        # needs sage.modules
sage: L = LazySymmetricFunctions(m)                                         # needs sage.modules
sage: L.one()                                                               # needs sage.modules
m[]

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> L.one()
1

>>> L = LazyPowerSeriesRing(ZZ, 'z')
>>> L.one()
1

>>> m = SymmetricFunctions(ZZ).m()                                        # needs sage.modules
>>> L = LazySymmetricFunctions(m)                                         # needs sage.modules
>>> L.one()                                                               # needs sage.modules
m[]

options = Current options for lazy series rings - constant_length: 3 - display_length: 7 - halting_precision: None - secure: False[source]#

prod(f, a=None, b=+Infinity, add_one=False)[source]#

The product of elements of self.

INPUT:

f – a list (or iterable) of elements of self
a, b – optional arguments
add_one – (default: False); if True, then converts a lazy series \(p_i\) from args into \(1 + p_i\) for the product

If a and b are both integers, then this returns the product \(\prod_{i=a}^b f(i)\), where \(f(i) = p_i\) if add_one=False or \(f(i) = 1 + p_i\) otherwise. If b is not specified, then we consider \(b = \infty\). Note this corresponds to the Python range(a, b+1).

If \(a\) is any other iterable, then this returns the product \(\prod_{i \in a} f(i)\), where \(f(i) = p_i\) if add_one=False or \(f(i) = 1 + p_i\).

Note

For infinite products, it is faster to use add_one=True since the implementation is based on \(p_i\) in \(\prod_i (1 + p_i)\).

Warning

When f is an infinite generator, then the first argument a must be True. Otherwise this will loop forever.

Warning

For an infinite product of the form \(\prod_i (1 + p_i)\), if \(p_i = 0\), then this will loop forever.

EXAMPLES:

Sage

sage: L.<t> = LazyLaurentSeriesRing(QQ)
sage: euler = L.prod(lambda n: 1 - t^n, PositiveIntegers())
sage: euler
1 - t - t^2 + t^5 + O(t^7)
sage: 1 / euler
1 + t + 2*t^2 + 3*t^3 + 5*t^4 + 7*t^5 + 11*t^6 + O(t^7)
sage: euler - L.euler()
O(t^7)
sage: L.prod(lambda n: -t^n, 1, add_one=True)
1 - t - t^2 + t^5 + O(t^7)

sage: L.prod((1 - t^n for n in PositiveIntegers()), True)
1 - t - t^2 + t^5 + O(t^7)
sage: L.prod((-t^n for n in PositiveIntegers()), True, add_one=True)
1 - t - t^2 + t^5 + O(t^7)

sage: L.prod((1 + t^(n-3) for n in PositiveIntegers()), True)
2*t^-3 + 4*t^-2 + 4*t^-1 + 4 + 6*t + 10*t^2 + 16*t^3 + O(t^4)

sage: L.prod(lambda n: 2 + t^n, -3, 5)
96*t^-6 + 240*t^-5 + 336*t^-4 + 840*t^-3 + 984*t^-2 + 1248*t^-1
 + 1980 + 1668*t + 1824*t^2 + 1872*t^3 + 1782*t^4 + 1710*t^5
 + 1314*t^6 + 1122*t^7 + 858*t^8 + 711*t^9 + 438*t^10 + 282*t^11
 + 210*t^12 + 84*t^13 + 60*t^14 + 24*t^15
sage: L.prod(lambda n: t^n / (1 + abs(n)), -2, 2, add_one=True)
1/3*t^-3 + 5/6*t^-2 + 13/9*t^-1 + 25/9 + 13/9*t + 5/6*t^2 + 1/3*t^3
sage: L.prod(lambda n: t^-2 + t^n / n, -4, -2)
1/24*t^-9 - 1/8*t^-8 - 1/6*t^-7 + 1/2*t^-6

sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: D.prod(lambda p: (1+D(1, valuation=p)).inverse(), Primes())
1 - 1/(2^s) - 1/(3^s) + 1/(4^s) - 1/(5^s) + 1/(6^s) - 1/(7^s) + O(1/(8^s))

sage: D.prod(lambda p: D(1, valuation=p), Primes(), add_one=True)
1 + 1/(2^s) + 1/(3^s) + 1/(5^s) + 1/(6^s) + 1/(7^s) + O(1/(8^s))

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('t',)); (t,) = L._first_ngens(1)
>>> euler = L.prod(lambda n: Integer(1) - t**n, PositiveIntegers())
>>> euler
1 - t - t^2 + t^5 + O(t^7)
>>> Integer(1) / euler
1 + t + 2*t^2 + 3*t^3 + 5*t^4 + 7*t^5 + 11*t^6 + O(t^7)
>>> euler - L.euler()
O(t^7)
>>> L.prod(lambda n: -t**n, Integer(1), add_one=True)
1 - t - t^2 + t^5 + O(t^7)

>>> L.prod((Integer(1) - t**n for n in PositiveIntegers()), True)
1 - t - t^2 + t^5 + O(t^7)
>>> L.prod((-t**n for n in PositiveIntegers()), True, add_one=True)
1 - t - t^2 + t^5 + O(t^7)

>>> L.prod((Integer(1) + t**(n-Integer(3)) for n in PositiveIntegers()), True)
2*t^-3 + 4*t^-2 + 4*t^-1 + 4 + 6*t + 10*t^2 + 16*t^3 + O(t^4)

>>> L.prod(lambda n: Integer(2) + t**n, -Integer(3), Integer(5))
96*t^-6 + 240*t^-5 + 336*t^-4 + 840*t^-3 + 984*t^-2 + 1248*t^-1
 + 1980 + 1668*t + 1824*t^2 + 1872*t^3 + 1782*t^4 + 1710*t^5
 + 1314*t^6 + 1122*t^7 + 858*t^8 + 711*t^9 + 438*t^10 + 282*t^11
 + 210*t^12 + 84*t^13 + 60*t^14 + 24*t^15
>>> L.prod(lambda n: t**n / (Integer(1) + abs(n)), -Integer(2), Integer(2), add_one=True)
1/3*t^-3 + 5/6*t^-2 + 13/9*t^-1 + 25/9 + 13/9*t + 5/6*t^2 + 1/3*t^3
>>> L.prod(lambda n: t**-Integer(2) + t**n / n, -Integer(4), -Integer(2))
1/24*t^-9 - 1/8*t^-8 - 1/6*t^-7 + 1/2*t^-6

>>> D = LazyDirichletSeriesRing(QQ, "s")
>>> D.prod(lambda p: (Integer(1)+D(Integer(1), valuation=p)).inverse(), Primes())
1 - 1/(2^s) - 1/(3^s) + 1/(4^s) - 1/(5^s) + 1/(6^s) - 1/(7^s) + O(1/(8^s))

>>> D.prod(lambda p: D(Integer(1), valuation=p), Primes(), add_one=True)
1 + 1/(2^s) + 1/(3^s) + 1/(5^s) + 1/(6^s) + 1/(7^s) + O(1/(8^s))

sum(f, a=None, b=+Infinity)[source]#

The sum of elements of self.

INPUT:

f – a list (or iterable or function) of elements of self
a, b – optional arguments

If a and b are both integers, then this returns the sum \(\sum_{i=a}^b f(i)\). If b is not specified, then we consider \(b = \infty\). Note this corresponds to the Python range(a, b+1).

If \(a\) is any other iterable, then this returns the sum \(\sum{i \in a} f(i)\).

Warning

When f is an infinite generator, then the first argument a must be True. Otherwise this will loop forever.

Warning

For an infinite sum of the form \(\sum_i s_i\), if \(s_i = 0\), then this will loop forever.

EXAMPLES:

Sage

sage: L.<t> = LazyLaurentSeriesRing(QQ)
sage: L.sum(lambda n: t^n / (n+1), PositiveIntegers())
1/2*t + 1/3*t^2 + 1/4*t^3 + 1/5*t^4 + 1/6*t^5 + 1/7*t^6 + 1/8*t^7 + O(t^8)

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: T = L.undefined(1)
sage: D = L.undefined(0)
sage: H = L.sum(lambda k: T(z^k)/k, 2)
sage: T.define(z*exp(T)*D)
sage: D.define(exp(H))
sage: T
z + z^2 + 2*z^3 + 4*z^4 + 9*z^5 + 20*z^6 + 48*z^7 + O(z^8)
sage: D
1 + 1/2*z^2 + 1/3*z^3 + 7/8*z^4 + 11/30*z^5 + 281/144*z^6 + O(z^7)

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('t',)); (t,) = L._first_ngens(1)
>>> L.sum(lambda n: t**n / (n+Integer(1)), PositiveIntegers())
1/2*t + 1/3*t^2 + 1/4*t^3 + 1/5*t^4 + 1/6*t^5 + 1/7*t^6 + 1/8*t^7 + O(t^8)

>>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> T = L.undefined(Integer(1))
>>> D = L.undefined(Integer(0))
>>> H = L.sum(lambda k: T(z**k)/k, Integer(2))
>>> T.define(z*exp(T)*D)
>>> D.define(exp(H))
>>> T
z + z^2 + 2*z^3 + 4*z^4 + 9*z^5 + 20*z^6 + 48*z^7 + O(z^8)
>>> D
1 + 1/2*z^2 + 1/3*z^3 + 7/8*z^4 + 11/30*z^5 + 281/144*z^6 + O(z^7)

We verify the Rogers-Ramanujan identities up to degree 100:

Sage

sage: L.<q> = LazyPowerSeriesRing(QQ)
sage: Gpi = L.prod(lambda k: -q^(1+5*k), 0, oo, add_one=True)
sage: Gpi *= L.prod(lambda k: -q^(4+5*k), 0, oo, add_one=True)
sage: Gp = 1 / Gpi
sage: G = L.sum(lambda n: q^(n^2) / prod(1 - q^(k+1) for k in range(n)), 0, oo)
sage: G - Gp
O(q^7)
sage: all(G[k] == Gp[k] for k in range(100))
True

sage: Hpi = L.prod(lambda k: -q^(2+5*k), 0, oo, add_one=True)
sage: Hpi *= L.prod(lambda k: -q^(3+5*k), 0, oo, add_one=True)
sage: Hp = 1 / Hpi
sage: H = L.sum(lambda n: q^(n^2+n) / prod(1 - q^(k+1) for k in range(n)), 0, oo)
sage: H - Hp
O(q^7)
sage: all(H[k] == Hp[k] for k in range(100))
True

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('q',)); (q,) = L._first_ngens(1)
>>> Gpi = L.prod(lambda k: -q**(Integer(1)+Integer(5)*k), Integer(0), oo, add_one=True)
>>> Gpi *= L.prod(lambda k: -q**(Integer(4)+Integer(5)*k), Integer(0), oo, add_one=True)
>>> Gp = Integer(1) / Gpi
>>> G = L.sum(lambda n: q**(n**Integer(2)) / prod(Integer(1) - q**(k+Integer(1)) for k in range(n)), Integer(0), oo)
>>> G - Gp
O(q^7)
>>> all(G[k] == Gp[k] for k in range(Integer(100)))
True

>>> Hpi = L.prod(lambda k: -q**(Integer(2)+Integer(5)*k), Integer(0), oo, add_one=True)
>>> Hpi *= L.prod(lambda k: -q**(Integer(3)+Integer(5)*k), Integer(0), oo, add_one=True)
>>> Hp = Integer(1) / Hpi
>>> H = L.sum(lambda n: q**(n**Integer(2)+n) / prod(Integer(1) - q**(k+Integer(1)) for k in range(n)), Integer(0), oo)
>>> H - Hp
O(q^7)
>>> all(H[k] == Hp[k] for k in range(Integer(100)))
True

Sage

sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: D.sum(lambda p: D(1, valuation=p), Primes())
1/(2^s) + 1/(3^s) + 1/(5^s) + 1/(7^s) + O(1/(9^s))

Python

>>> from sage.all import *
>>> D = LazyDirichletSeriesRing(QQ, "s")
>>> D.sum(lambda p: D(Integer(1), valuation=p), Primes())
1/(2^s) + 1/(3^s) + 1/(5^s) + 1/(7^s) + O(1/(9^s))

undefined(valuation=None)[source]#

Return an uninitialized series.

INPUT:

valuation – integer; a lower bound for the valuation of the series

Power series can be defined recursively (see sage.rings.lazy_series.LazyModuleElement.define() for more examples).

EXAMPLES:

Sage

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: s = L.undefined(1)
sage: s.define(z + (s^2+s(z^2))/2)
sage: s
z + z^2 + z^3 + 2*z^4 + 3*z^5 + 6*z^6 + 11*z^7 + O(z^8)

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> s = L.undefined(Integer(1))
>>> s.define(z + (s**Integer(2)+s(z**Integer(2)))/Integer(2))
>>> s
z + z^2 + z^3 + 2*z^4 + 3*z^5 + 6*z^6 + 11*z^7 + O(z^8)

Alternatively:

Sage

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = L(None, valuation=-1)
sage: f.define(z^-1 + z^2*f^2)
sage: f
z^-1 + 1 + 2*z + 5*z^2 + 14*z^3 + 42*z^4 + 132*z^5 + O(z^6)

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = L(None, valuation=-Integer(1))
>>> f.define(z**-Integer(1) + z**Integer(2)*f**Integer(2))
>>> f
z^-1 + 1 + 2*z + 5*z^2 + 14*z^3 + 42*z^4 + 132*z^5 + O(z^6)

unknown(valuation=None)[source]#

Return an uninitialized series.

INPUT:

valuation – integer; a lower bound for the valuation of the series

Power series can be defined recursively (see sage.rings.lazy_series.LazyModuleElement.define() for more examples).

EXAMPLES:

Sage

sage: L.<z> = LazyPowerSeriesRing(QQ)
sage: s = L.undefined(1)
sage: s.define(z + (s^2+s(z^2))/2)
sage: s
z + z^2 + z^3 + 2*z^4 + 3*z^5 + 6*z^6 + 11*z^7 + O(z^8)

Python

>>> from sage.all import *
>>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> s = L.undefined(Integer(1))
>>> s.define(z + (s**Integer(2)+s(z**Integer(2)))/Integer(2))
>>> s
z + z^2 + z^3 + 2*z^4 + 3*z^5 + 6*z^6 + 11*z^7 + O(z^8)

Alternatively:

Sage

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = L(None, valuation=-1)
sage: f.define(z^-1 + z^2*f^2)
sage: f
z^-1 + 1 + 2*z + 5*z^2 + 14*z^3 + 42*z^4 + 132*z^5 + O(z^6)

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)
>>> f = L(None, valuation=-Integer(1))
>>> f.define(z**-Integer(1) + z**Integer(2)*f**Integer(2))
>>> f
z^-1 + 1 + 2*z + 5*z^2 + 14*z^3 + 42*z^4 + 132*z^5 + O(z^6)

zero()[source]#

Return the zero series.

EXAMPLES:

Sage

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: L.zero()
0

sage: s = SymmetricFunctions(ZZ).s()                                        # needs sage.modules
sage: L = LazySymmetricFunctions(s)                                         # needs sage.modules
sage: L.zero()                                                              # needs sage.modules
0

sage: L = LazyDirichletSeriesRing(ZZ, 'z')
sage: L.zero()
0

sage: L = LazyPowerSeriesRing(ZZ, 'z')
sage: L.zero()
0

Python

>>> from sage.all import *
>>> L = LazyLaurentSeriesRing(ZZ, 'z')
>>> L.zero()
0

>>> s = SymmetricFunctions(ZZ).s()                                        # needs sage.modules
>>> L = LazySymmetricFunctions(s)                                         # needs sage.modules
>>> L.zero()                                                              # needs sage.modules
0

>>> L = LazyDirichletSeriesRing(ZZ, 'z')
>>> L.zero()
0

>>> L = LazyPowerSeriesRing(ZZ, 'z')
>>> L.zero()
0

class sage.rings.lazy_series_ring.LazySymmetricFunctions(basis, sparse=True, category=None)[source]#

Bases: LazyCompletionGradedAlgebra

The ring of lazy symmetric functions.

INPUT:

basis – the ring of symmetric functions
names – name(s) of the alphabets
sparse – (default: True) whether we use a sparse or a dense representation

EXAMPLES:

Sage

sage: s = SymmetricFunctions(ZZ).s()                                            # needs sage.modules
sage: LazySymmetricFunctions(s)                                                 # needs sage.modules
Lazy completion of Symmetric Functions over Integer Ring in the Schur basis

sage: m = SymmetricFunctions(ZZ).m()                                            # needs sage.modules
sage: LazySymmetricFunctions(tensor([s, m]))                                    # needs sage.modules
Lazy completion of
 Symmetric Functions over Integer Ring in the Schur basis
  # Symmetric Functions over Integer Ring in the monomial basis

Python

>>> from sage.all import *
>>> s = SymmetricFunctions(ZZ).s()                                            # needs sage.modules
>>> LazySymmetricFunctions(s)                                                 # needs sage.modules
Lazy completion of Symmetric Functions over Integer Ring in the Schur basis

>>> m = SymmetricFunctions(ZZ).m()                                            # needs sage.modules
>>> LazySymmetricFunctions(tensor([s, m]))                                    # needs sage.modules
Lazy completion of
 Symmetric Functions over Integer Ring in the Schur basis
  # Symmetric Functions over Integer Ring in the monomial basis

Element[source]#: alias of LazySymmetricFunction