Lazy Series Rings¶
We provide lazy implementations for various \(\NN\)-graded rings.
| The ring of lazy Laurent series. | |
| The ring of (possibly multivariate) lazy Taylor series. | |
| The completion of a graded algebra consisting of formal series. | |
| The ring of (possibly multivariate) lazy symmetric functions. | |
| 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: L.<x> = LazyPowerSeriesRing(ZZ)
sage: f = L(lambda n: 0, valuation=0)
sage: 1 / f  # not tested - infinite loop
>>> 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– boolean (default:- True); whether we use a sparse or a dense representation
 - EXAMPLES: - 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 - >>> 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: 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])] - >>> 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– boolean (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: LazyDirichletSeriesRing(ZZ, 't') Lazy Dirichlet Series Ring in t over Integer Ring - >>> 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: L = LazyDirichletSeriesRing(QQ, 's') sage: L in PrincipalIdealDomains False - >>> 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: L = LazyDirichletSeriesRing(ZZ, 'z') sage: L.one() # needs sage.symbolic 1 sage: ~L.one() # needs sage.symbolic 1 + O(1/(8^z)) - >>> 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: 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))] - >>> 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– boolean (default:- True); whether the implementation of the series is sparse or not
 - EXAMPLES: - 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 - >>> 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: # 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 - >>> 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: 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) - >>> 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: 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) - >>> 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: 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) - >>> 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: 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) - >>> 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: 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) - >>> 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 - fand- gthat are not known to be equal are considered to be different:- 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 - >>> 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 != greturning- Truemight not mean that the two series are actually different:- sage: f = L(lambda n: 0, valuation=0) sage: g = L.zero() sage: f != g True - >>> 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- Trueif the series is known to be nonzero:- sage: (f - g).is_nonzero() False - >>> 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: L.<z> = LazyLaurentSeriesRing(ZZ) sage: L.is_sparse() True sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False) sage: L.is_sparse() False - >>> 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 - ValueErrorand the second uses a check up to a (user set) finite precision. These behaviors are set using the options- secureand- 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.secureis set to- True, then we raise a- ValueError:- 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 - >>> 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: bool(zf) Traceback (most recent call last): ... ValueError: undecidable - >>> 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: 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 - >>> 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: L.<q> = LazyLaurentSeriesRing(ZZ) sage: phi = q.euler() sage: phi 1 - q - q^2 + q^5 + O(q^7) - >>> 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: 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 - >>> 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: 
 - gen(n=0)[source]¶
- Return the - n-th generator of- self.- EXAMPLES: - sage: L = LazyLaurentSeriesRing(ZZ, 'z') sage: L.gen() z sage: L.gen(3) Traceback (most recent call last): ... IndexError: there is only one generator - >>> 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: L.<z> = LazyLaurentSeriesRing(ZZ) sage: L.gens() (z,) sage: 1/(1 - z) 1 + z + z^2 + O(z^3) - >>> 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: L.<z> = LazyLaurentSeriesRing(ZZ) sage: L.ngens() 1 - >>> 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: 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) - >>> 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: 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) - >>> 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: 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 - >>> 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: L = LazyLaurentSeriesRing(QQ, 'z') sage: L.residue_field() Rational Field - >>> 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 - coefficientreturns the value of the coefficient of index \(i\) from input \(s\) and \(i\) where \(s\) is the series itself.- Let - valuationbe \(n\). All coefficients of index below \(n\) are zero. If- constantis not specified, then the- coefficientfunction is responsible to compute the values of all coefficients of index \(\ge n\). If- degreeor- constantis a pair \((c,m)\), then the- coefficientfunction 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: 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 - >>> 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 - coefficientcan 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,- constantmay be just an element of the base ring instead of a tuple or can be simply omitted if it is zero.- 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) - >>> 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: 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)] - >>> 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 - derivativemethod
 
 - EXAMPLES: - 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) - >>> 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: 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) - >>> 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) 
 
- 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– boolean (default:- True); whether this series is sparse or not
 - EXAMPLES: - 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 - >>> 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
 - construction()[source]¶
- Return a pair - (F, R), where- Fis a- CompletionFunctorand \(R\) is a ring, such that- F(R)returns- self.- EXAMPLES: - sage: L = LazyPowerSeriesRing(ZZ, 't') sage: L.construction() (Completion[t, prec=+Infinity], Sparse Univariate Polynomial Ring in t over Integer Ring) - >>> from sage.all import * >>> L = LazyPowerSeriesRing(ZZ, 't') >>> L.construction() (Completion[t, prec=+Infinity], Sparse Univariate Polynomial Ring in t over Integer Ring) 
 - 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: L.<x> = LazyPowerSeriesRing(QQ) sage: L.fraction_field() Lazy Laurent Series Ring in x over Rational Field - >>> 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: L = LazyPowerSeriesRing(ZZ, 'z') sage: L.gen() z sage: L.gen(3) Traceback (most recent call last): ... IndexError: there is only one generator - >>> 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: L = LazyPowerSeriesRing(ZZ, 'x,y') sage: L.gens() (x, y) - >>> from sage.all import * >>> L = LazyPowerSeriesRing(ZZ, 'x,y') >>> L.gens() (x, y) 
 - ngens()[source]¶
- Return the number of generators of - self.- EXAMPLES: - sage: L.<z> = LazyPowerSeriesRing(ZZ) sage: L.ngens() 1 - >>> 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: L = LazyPowerSeriesRing(QQ, 'x') sage: L.residue_field() Rational Field - >>> 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: 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] - >>> 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 - derivativemethod
 
 - Warning - For inputs as symbolic functions/expressions, this does not check that the function does not have poles at \(0\). - EXAMPLES: - 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) - >>> 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: L.<z> = LazyPowerSeriesRing(QQ, sparse=True) sage: f = 1 / sin(x) sage: L.taylor(f) <repr(...) failed: ValueError: power::eval(): division by zero> - >>> 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: 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 - >>> 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 
 
- 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: 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 - >>> 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 
 - define_implicitly(series, equations, max_lookahead=1)[source]¶
- Define series by solving functional equations. - INPUT: - series– list of undefined series or pairs each consisting of a series and its initial values
- equations– list of equations defining the series
- max_lookahead– (default:- 1); a positive integer specifying how many elements beyond the currently known (i.e., approximate) order of each equation to extract linear equations from
 - EXAMPLES: - sage: L.<z> = LazyPowerSeriesRing(QQ) sage: f = L.undefined(0) sage: F = diff(f, 2) sage: L.define_implicitly([(f, [1, 0])], [F + f]) sage: f 1 - 1/2*z^2 + 1/24*z^4 - 1/720*z^6 + O(z^7) sage: cos(z) 1 - 1/2*z^2 + 1/24*z^4 - 1/720*z^6 + O(z^7) sage: F -1 + 1/2*z^2 - 1/24*z^4 + 1/720*z^6 + O(z^7) sage: L.<z> = LazyPowerSeriesRing(QQ) sage: f = L.undefined(0) sage: L.define_implicitly([f], [2*z*f(z^3) + z*f^3 - 3*f + 3]) sage: f 1 + z + z^2 + 2*z^3 + 5*z^4 + 11*z^5 + 28*z^6 + O(z^7) - >>> from sage.all import * >>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1) >>> f = L.undefined(Integer(0)) >>> F = diff(f, Integer(2)) >>> L.define_implicitly([(f, [Integer(1), Integer(0)])], [F + f]) >>> f 1 - 1/2*z^2 + 1/24*z^4 - 1/720*z^6 + O(z^7) >>> cos(z) 1 - 1/2*z^2 + 1/24*z^4 - 1/720*z^6 + O(z^7) >>> F -1 + 1/2*z^2 - 1/24*z^4 + 1/720*z^6 + O(z^7) >>> L = LazyPowerSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1) >>> f = L.undefined(Integer(0)) >>> L.define_implicitly([f], [Integer(2)*z*f(z**Integer(3)) + z*f**Integer(3) - Integer(3)*f + Integer(3)]) >>> f 1 + z + z^2 + 2*z^3 + 5*z^4 + 11*z^5 + 28*z^6 + O(z^7) - From Exercise 6.63b in [EnumComb2]: - sage: g = L.undefined() sage: z1 = z*diff(g, z) sage: z2 = z1 + z^2 * diff(g, z, 2) sage: z3 = z1 + 3 * z^2 * diff(g, z, 2) + z^3 * diff(g, z, 3) sage: e1 = g^2 * z3 - 15*g*z1*z2 + 30*z1^3 sage: e2 = g * z2 - 3 * z1^2 sage: e3 = g * z2 - 3 * z1^2 sage: e = e1^2 + 32 * e2^3 - g^10 * e3^2 sage: L.define_implicitly([(g, [1, 2])], [e]) sage: sol = L(lambda n: 1 if not n else (2 if is_square(n) else 0)); sol 1 + 2*z + 2*z^4 + O(z^7) sage: all(g[i] == sol[i] for i in range(50)) True - >>> from sage.all import * >>> g = L.undefined() >>> z1 = z*diff(g, z) >>> z2 = z1 + z**Integer(2) * diff(g, z, Integer(2)) >>> z3 = z1 + Integer(3) * z**Integer(2) * diff(g, z, Integer(2)) + z**Integer(3) * diff(g, z, Integer(3)) >>> e1 = g**Integer(2) * z3 - Integer(15)*g*z1*z2 + Integer(30)*z1**Integer(3) >>> e2 = g * z2 - Integer(3) * z1**Integer(2) >>> e3 = g * z2 - Integer(3) * z1**Integer(2) >>> e = e1**Integer(2) + Integer(32) * e2**Integer(3) - g**Integer(10) * e3**Integer(2) >>> L.define_implicitly([(g, [Integer(1), Integer(2)])], [e]) >>> sol = L(lambda n: Integer(1) if not n else (Integer(2) if is_square(n) else Integer(0))); sol 1 + 2*z + 2*z^4 + O(z^7) >>> all(g[i] == sol[i] for i in range(Integer(50))) True - Some more examples over different rings: - sage: # needs sage.symbolic sage: L.<z> = LazyPowerSeriesRing(SR) sage: G = L.undefined(0) sage: L.define_implicitly([(G, [ln(2)])], [diff(G) - exp(-G(-z))]) sage: G log(2) + z + 1/2*z^2 + (-1/12*z^4) + 1/45*z^6 + O(z^7) sage: L.<z> = LazyPowerSeriesRing(RR) sage: G = L.undefined(0) sage: L.define_implicitly([(G, [log(2)])], [diff(G) - exp(-G(-z))]) sage: G 0.693147180559945 + 1.00000000000000*z + 0.500000000000000*z^2 - 0.0833333333333333*z^4 + 0.0222222222222222*z^6 + O(1.00000000000000*z^7) - >>> from sage.all import * >>> # needs sage.symbolic >>> L = LazyPowerSeriesRing(SR, names=('z',)); (z,) = L._first_ngens(1) >>> G = L.undefined(Integer(0)) >>> L.define_implicitly([(G, [ln(Integer(2))])], [diff(G) - exp(-G(-z))]) >>> G log(2) + z + 1/2*z^2 + (-1/12*z^4) + 1/45*z^6 + O(z^7) >>> L = LazyPowerSeriesRing(RR, names=('z',)); (z,) = L._first_ngens(1) >>> G = L.undefined(Integer(0)) >>> L.define_implicitly([(G, [log(Integer(2))])], [diff(G) - exp(-G(-z))]) >>> G 0.693147180559945 + 1.00000000000000*z + 0.500000000000000*z^2 - 0.0833333333333333*z^4 + 0.0222222222222222*z^6 + O(1.00000000000000*z^7) - We solve the recurrence relation in (3.12) of Prellberg and Brak doi:10.1007/BF02183685: - sage: q, y = QQ['q,y'].fraction_field().gens() sage: L.<x> = LazyPowerSeriesRing(q.parent()) sage: R = L.undefined() sage: L.define_implicitly([(R, [0])], [(1-q*x)*R - (y*q*x+y)*R(q*x) - q*x*R*R(q*x) - x*y*q]) sage: R[0] 0 sage: R[1] (-q*y)/(q*y - 1) sage: R[2] (q^3*y^2 + q^2*y)/(q^3*y^2 - q^2*y - q*y + 1) sage: R[3].factor() (-1) * y * q^3 * (q*y - 1)^-2 * (q^2*y - 1)^-1 * (q^3*y - 1)^-1 * (q^4*y^3 + q^3*y^2 + q^2*y^2 - q^2*y - q*y - 1) sage: Rp = L.undefined(1) sage: L.define_implicitly([Rp], [(y*q*x+y)*Rp(q*x) + q*x*Rp*Rp(q*x) + x*y*q - (1-q*x)*Rp]) sage: all(R[n] == Rp[n] for n in range(7)) True - >>> from sage.all import * >>> q, y = QQ['q,y'].fraction_field().gens() >>> L = LazyPowerSeriesRing(q.parent(), names=('x',)); (x,) = L._first_ngens(1) >>> R = L.undefined() >>> L.define_implicitly([(R, [Integer(0)])], [(Integer(1)-q*x)*R - (y*q*x+y)*R(q*x) - q*x*R*R(q*x) - x*y*q]) >>> R[Integer(0)] 0 >>> R[Integer(1)] (-q*y)/(q*y - 1) >>> R[Integer(2)] (q^3*y^2 + q^2*y)/(q^3*y^2 - q^2*y - q*y + 1) >>> R[Integer(3)].factor() (-1) * y * q^3 * (q*y - 1)^-2 * (q^2*y - 1)^-1 * (q^3*y - 1)^-1 * (q^4*y^3 + q^3*y^2 + q^2*y^2 - q^2*y - q*y - 1) >>> Rp = L.undefined(Integer(1)) >>> L.define_implicitly([Rp], [(y*q*x+y)*Rp(q*x) + q*x*Rp*Rp(q*x) + x*y*q - (Integer(1)-q*x)*Rp]) >>> all(R[n] == Rp[n] for n in range(Integer(7))) True - Another example: - sage: L.<z> = LazyPowerSeriesRing(QQ["x,y,f1,f2"].fraction_field()) sage: L.base_ring().inject_variables() Defining x, y, f1, f2 sage: F = L.undefined() sage: L.define_implicitly([(F, [0, f1, f2])], [F(2*z) - (1+exp(x*z)+exp(y*z))*F - exp((x+y)*z)*F(-z)]) sage: F f1*z + f2*z^2 + ((-1/6*x*y*f1+1/3*x*f2+1/3*y*f2)*z^3) + ((-1/24*x^2*y*f1-1/24*x*y^2*f1+1/12*x^2*f2+1/12*x*y*f2+1/12*y^2*f2)*z^4) + ... + O(z^8) sage: sol = 1/(x-y)*((2*f2-y*f1)*(exp(x*z)-1)/x - (2*f2-x*f1)*(exp(y*z)-1)/y) sage: F - sol O(z^7) - >>> from sage.all import * >>> L = LazyPowerSeriesRing(QQ["x,y,f1,f2"].fraction_field(), names=('z',)); (z,) = L._first_ngens(1) >>> L.base_ring().inject_variables() Defining x, y, f1, f2 >>> F = L.undefined() >>> L.define_implicitly([(F, [Integer(0), f1, f2])], [F(Integer(2)*z) - (Integer(1)+exp(x*z)+exp(y*z))*F - exp((x+y)*z)*F(-z)]) >>> F f1*z + f2*z^2 + ((-1/6*x*y*f1+1/3*x*f2+1/3*y*f2)*z^3) + ((-1/24*x^2*y*f1-1/24*x*y^2*f1+1/12*x^2*f2+1/12*x*y*f2+1/12*y^2*f2)*z^4) + ... + O(z^8) >>> sol = Integer(1)/(x-y)*((Integer(2)*f2-y*f1)*(exp(x*z)-Integer(1))/x - (Integer(2)*f2-x*f1)*(exp(y*z)-Integer(1))/y) >>> F - sol O(z^7) - We need to specify the initial values for the degree 1 and 2 components to get a unique solution in the previous example: - sage: L.<z> = LazyPowerSeriesRing(QQ['x','y','f1'].fraction_field()) sage: L.base_ring().inject_variables() Defining x, y, f1 sage: F = L.undefined() sage: L.define_implicitly([F], [F(2*z) - (1+exp(x*z)+exp(y*z))*F - exp((x+y)*z)*F(-z)]) sage: F <repr(...) failed: ValueError: could not determine any coefficients: coefficient [3]: 6*series[3] + (-2*x - 2*y)*series[2] + (x*y)*series[1] == 0> - >>> from sage.all import * >>> L = LazyPowerSeriesRing(QQ['x','y','f1'].fraction_field(), names=('z',)); (z,) = L._first_ngens(1) >>> L.base_ring().inject_variables() Defining x, y, f1 >>> F = L.undefined() >>> L.define_implicitly([F], [F(Integer(2)*z) - (Integer(1)+exp(x*z)+exp(y*z))*F - exp((x+y)*z)*F(-z)]) >>> F <repr(...) failed: ValueError: could not determine any coefficients: coefficient [3]: 6*series[3] + (-2*x - 2*y)*series[2] + (x*y)*series[1] == 0> - Let us now try to only specify the degree 0 and degree 1 components. We will see that this is still not enough to remove the ambiguity, so an error is raised. However, we will see that the dependence on - series[1]disappears. The equation which has no unique solution is now- 6*series[3] + (-2*x - 2*y)*series[2] + (x*y*f1) == 0.:- sage: F = L.undefined() sage: L.define_implicitly([(F, [0, f1])], [F(2*z) - (1+exp(x*z)+exp(y*z))*F - exp((x+y)*z)*F(-z)]) sage: F <repr(...) failed: ValueError: could not determine any coefficients: coefficient [3]: ... == 0> - >>> from sage.all import * >>> F = L.undefined() >>> L.define_implicitly([(F, [Integer(0), f1])], [F(Integer(2)*z) - (Integer(1)+exp(x*z)+exp(y*z))*F - exp((x+y)*z)*F(-z)]) >>> F <repr(...) failed: ValueError: could not determine any coefficients: coefficient [3]: ... == 0> - (Note that the order of summands of the equation in the error message is not deterministic.) - Laurent series examples: - sage: L.<z> = LazyLaurentSeriesRing(QQ) sage: f = L.undefined(-1) sage: L.define_implicitly([(f, [5])], [2+z*f(z^2) - f]) sage: f 5*z^-1 + 2 + 2*z + 2*z^3 + O(z^6) sage: 2 + z*f(z^2) - f O(z^6) sage: g = L.undefined(-2) sage: L.define_implicitly([(g, [5])], [2+z*g(z^2) - g]) sage: g <repr(...) failed: ValueError: no solution as the coefficient in degree -3 of the equation is 5 != 0> - >>> from sage.all import * >>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1) >>> f = L.undefined(-Integer(1)) >>> L.define_implicitly([(f, [Integer(5)])], [Integer(2)+z*f(z**Integer(2)) - f]) >>> f 5*z^-1 + 2 + 2*z + 2*z^3 + O(z^6) >>> Integer(2) + z*f(z**Integer(2)) - f O(z^6) >>> g = L.undefined(-Integer(2)) >>> L.define_implicitly([(g, [Integer(5)])], [Integer(2)+z*g(z**Integer(2)) - g]) >>> g <repr(...) failed: ValueError: no solution as the coefficient in degree -3 of the equation is 5 != 0> - A bivariate example: - sage: L.<x, y> = LazyPowerSeriesRing(QQ) sage: B = L.undefined() sage: eq = y*B^2 + 1 - B(x, x-y) sage: L.define_implicitly([B], [eq]) sage: B 1 + (x-y) + (2*x*y-2*y^2) + (4*x^2*y-7*x*y^2+3*y^3) + (2*x^3*y+6*x^2*y^2-18*x*y^3+10*y^4) + (30*x^3*y^2-78*x^2*y^3+66*x*y^4-18*y^5) + (28*x^4*y^2-12*x^3*y^3-128*x^2*y^4+180*x*y^5-68*y^6) + O(x,y)^7 - >>> from sage.all import * >>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2) >>> B = L.undefined() >>> eq = y*B**Integer(2) + Integer(1) - B(x, x-y) >>> L.define_implicitly([B], [eq]) >>> B 1 + (x-y) + (2*x*y-2*y^2) + (4*x^2*y-7*x*y^2+3*y^3) + (2*x^3*y+6*x^2*y^2-18*x*y^3+10*y^4) + (30*x^3*y^2-78*x^2*y^3+66*x*y^4-18*y^5) + (28*x^4*y^2-12*x^3*y^3-128*x^2*y^4+180*x*y^5-68*y^6) + O(x,y)^7 - Knödel walks: - sage: L.<z, x> = LazyPowerSeriesRing(QQ) sage: F = L.undefined() sage: eq = F(z, x)*(x^2*z-x+z) - (z - x*z^2 - x^2*z^2)*F(z, 0) + x sage: L.define_implicitly([F], [eq]) sage: F 1 + (2*z^2+z*x) + (z^3+z^2*x) + (5*z^4+3*z^3*x+z^2*x^2) + (5*z^5+4*z^4*x+z^3*x^2) + (15*z^6+10*z^5*x+4*z^4*x^2+z^3*x^3) + O(z,x)^7 - >>> from sage.all import * >>> L = LazyPowerSeriesRing(QQ, names=('z', 'x',)); (z, x,) = L._first_ngens(2) >>> F = L.undefined() >>> eq = F(z, x)*(x**Integer(2)*z-x+z) - (z - x*z**Integer(2) - x**Integer(2)*z**Integer(2))*F(z, Integer(0)) + x >>> L.define_implicitly([F], [eq]) >>> F 1 + (2*z^2+z*x) + (z^3+z^2*x) + (5*z^4+3*z^3*x+z^2*x^2) + (5*z^5+4*z^4*x+z^3*x^2) + (15*z^6+10*z^5*x+4*z^4*x^2+z^3*x^3) + O(z,x)^7 - Bicolored rooted trees with black and white roots: - sage: L.<x, y> = LazyPowerSeriesRing(QQ) sage: A = L.undefined() sage: B = L.undefined() sage: L.define_implicitly([A, B], [A - x*exp(B), B - y*exp(A)]) sage: A x + x*y + (x^2*y+1/2*x*y^2) + (1/2*x^3*y+2*x^2*y^2+1/6*x*y^3) + (1/6*x^4*y+3*x^3*y^2+2*x^2*y^3+1/24*x*y^4) + (1/24*x^5*y+8/3*x^4*y^2+27/4*x^3*y^3+4/3*x^2*y^4+1/120*x*y^5) + O(x,y)^7 sage: h = SymmetricFunctions(QQ).h() sage: S = LazySymmetricFunctions(h) sage: E = S(lambda n: h[n]) sage: T = LazySymmetricFunctions(tensor([h, h])) sage: X = tensor([h[1],h[[]]]) sage: Y = tensor([h[[]],h[1]]) sage: A = T.undefined() sage: B = T.undefined() sage: T.define_implicitly([A, B], [A - X*E(B), B - Y*E(A)]) sage: A[:3] [h[1] # h[], h[1] # h[1]] - >>> from sage.all import * >>> L = LazyPowerSeriesRing(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2) >>> A = L.undefined() >>> B = L.undefined() >>> L.define_implicitly([A, B], [A - x*exp(B), B - y*exp(A)]) >>> A x + x*y + (x^2*y+1/2*x*y^2) + (1/2*x^3*y+2*x^2*y^2+1/6*x*y^3) + (1/6*x^4*y+3*x^3*y^2+2*x^2*y^3+1/24*x*y^4) + (1/24*x^5*y+8/3*x^4*y^2+27/4*x^3*y^3+4/3*x^2*y^4+1/120*x*y^5) + O(x,y)^7 >>> h = SymmetricFunctions(QQ).h() >>> S = LazySymmetricFunctions(h) >>> E = S(lambda n: h[n]) >>> T = LazySymmetricFunctions(tensor([h, h])) >>> X = tensor([h[Integer(1)],h[[]]]) >>> Y = tensor([h[[]],h[Integer(1)]]) >>> A = T.undefined() >>> B = T.undefined() >>> T.define_implicitly([A, B], [A - X*E(B), B - Y*E(A)]) >>> A[:Integer(3)] [h[1] # h[], h[1] # h[1]] - Permutations with two kinds of labels such that each cycle contains at least one element of each kind (defined implicitly to have a test): - sage: p = SymmetricFunctions(QQ).p() sage: S = LazySymmetricFunctions(p) sage: P = S(lambda n: sum(p[la] for la in Partitions(n))) sage: T = LazySymmetricFunctions(tensor([p, p])) sage: X = tensor([p[1],p[[]]]) sage: Y = tensor([p[[]],p[1]]) sage: A = T.undefined() sage: T.define_implicitly([A], [P(X)*P(Y)*A - P(X+Y)]) sage: A[:4] [p[] # p[], 0, p[1] # p[1], p[1] # p[1, 1] + p[1, 1] # p[1]] - >>> from sage.all import * >>> p = SymmetricFunctions(QQ).p() >>> S = LazySymmetricFunctions(p) >>> P = S(lambda n: sum(p[la] for la in Partitions(n))) >>> T = LazySymmetricFunctions(tensor([p, p])) >>> X = tensor([p[Integer(1)],p[[]]]) >>> Y = tensor([p[[]],p[Integer(1)]]) >>> A = T.undefined() >>> T.define_implicitly([A], [P(X)*P(Y)*A - P(X+Y)]) >>> A[:Integer(4)] [p[] # p[], 0, p[1] # p[1], p[1] # p[1, 1] + p[1, 1] # p[1]] - The Frobenius character of labelled Dyck words: - sage: h = SymmetricFunctions(QQ).h() sage: L.<t, u> = LazyPowerSeriesRing(h.fraction_field()) sage: D = L.undefined() sage: s1 = L.sum(lambda n: h[n]*t^(n+1)*u^(n-1), 1) sage: L.define_implicitly([D], [u*D - u - u*s1*D - t*(D - D(t, 0))]) sage: D h[] + h[1]*t^2 + ((h[1,1]+h[2])*t^4+h[2]*t^3*u) + ((h[1,1,1]+3*h[2,1]+h[3])*t^6+(2*h[2,1]+h[3])*t^5*u+h[3]*t^4*u^2) + O(t,u)^7 - >>> from sage.all import * >>> h = SymmetricFunctions(QQ).h() >>> L = LazyPowerSeriesRing(h.fraction_field(), names=('t', 'u',)); (t, u,) = L._first_ngens(2) >>> D = L.undefined() >>> s1 = L.sum(lambda n: h[n]*t**(n+Integer(1))*u**(n-Integer(1)), Integer(1)) >>> L.define_implicitly([D], [u*D - u - u*s1*D - t*(D - D(t, Integer(0)))]) >>> D h[] + h[1]*t^2 + ((h[1,1]+h[2])*t^4+h[2]*t^3*u) + ((h[1,1,1]+3*h[2,1]+h[3])*t^6+(2*h[2,1]+h[3])*t^5*u+h[3]*t^4*u^2) + O(t,u)^7 
 - is_exact()[source]¶
- Return if - selfis exact or not.- EXAMPLES: - sage: L = LazyLaurentSeriesRing(ZZ, 'z') sage: L.is_exact() True sage: L = LazyLaurentSeriesRing(RR, 'z') sage: L.is_exact() False - >>> 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 - selfis sparse or not.- EXAMPLES: - sage: L = LazyLaurentSeriesRing(ZZ, 'z', sparse=False) sage: L.is_sparse() False sage: L = LazyLaurentSeriesRing(ZZ, 'z', sparse=True) sage: L.is_sparse() True - >>> 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: 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[] - >>> 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– 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- argsinto \(1 + p_i\) for the product
 - If - aand- bare both integers, then this returns the product \(\prod_{i=a}^b f(i)\), where \(f(i) = p_i\) if- add_one=Falseor \(f(i) = 1 + p_i\) otherwise. If- bis 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=Falseor \(f(i) = 1 + p_i\).- Note - For infinite products, it is faster to use - add_one=Truesince the implementation is based on \(p_i\) in \(\prod_i (1 + p_i)\).- Warning - When - fis an infinite generator, then the first argument- amust 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: 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)) - >>> 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– list (or iterable or function) of elements of- self
- a,- b– optional arguments
 - If - aand- bare both integers, then this returns the sum \(\sum_{i=a}^b f(i)\). If- bis 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 - fis an infinite generator, then the first argument- amust 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: 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) - >>> 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: 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 - >>> 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: 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)) - >>> 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, name=None)[source]¶
- Return an uninitialized series. - INPUT: - valuation– integer; a lower bound for the valuation of the series
- name– string; a name that refers to the undefined stream in error messages
 - Power series can be defined recursively (see - sage.rings.lazy_series.LazyModuleElement.define()for more examples).- EXAMPLES: - 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) - >>> 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: 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) - >>> 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, name=None)[source]¶
- Return an uninitialized series. - INPUT: - valuation– integer; a lower bound for the valuation of the series
- name– string; a name that refers to the undefined stream in error messages
 - Power series can be defined recursively (see - sage.rings.lazy_series.LazyModuleElement.define()for more examples).- EXAMPLES: - 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) - >>> 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: 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) - >>> 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: 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 - >>> 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– boolean (default:- True); whether we use a sparse or a dense representation
 - EXAMPLES: - 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 - >>> 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