Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

Schneider, Carsten

doi:10.48550/arxiv.2102.01471

Cited by 4 publications

(9 citation statements)

References 116 publications

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“…In general, multi-sums appear with complicated hypergeometric products and one may try to apply, e.g., the package EvaluateMultiSums [36][37][38][39] (utilizing the difference ring algorithms [42][43][44] available in Sigma) to represent these sums to indefinite nested sums. In general, this seems not possible.…”

Section: Computing the Expansion In εmentioning

confidence: 99%

“…In practice, it is important for this list of nested sums to be shiftstable, meaning that a shift in any of the variables must not introduce new harmonic sums not already included in the list, and they also should be linearly independent. To guarantee this property, one can use quasi-shuffle algebras or difference ring methods [43,73]. The nested sums at shifted arguments can then be rewritten through the repeated application of identities of the type…”

Section: Finding Solutions In Terms Of Nested Sumsmentioning

confidence: 99%

“…One may try to simplify the obtained multiple sums in terms of hypergeometric products and indefinite nested sums to expressions that are purely given in terms of indefinite nested sums using the package EvaluateMultiSums [36][37][38][39]. The underlying summation engine is based on the package Sigma [40,41] that contains non-trivial algorithms in the setting of difference rings [42][43][44].…”

Section: Introductionmentioning

confidence: 99%

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Hypergeometric Structures in Feynman Integrals

Blümlein¹,

Saragnese²,

Schneider³

2021

Preprint

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Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton-type functions are considered. We illustrate the algorithms by examples.

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Section: Computing the Expansion In εmentioning

confidence: 99%

Section: Finding Solutions In Terms Of Nested Sumsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hypergeometric Structures in Feynman Integrals

Blümlein¹,

Saragnese²,

Schneider³

2021

Preprint

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“…For the ordinary case r = 1 the obtained recurrences can be solved in terms of indefinite nested sums defined over hypergeometric products 1 using difference ring techniques [61][62][63][64][65][66][67]. This leads to a general method that can find all solutions of a given univariate linear differential equation in terms of power series solutions where the expansion coefficients are given in terms of hypergeometric products or indefinite nested sums defined over such products.…”

Section: Introductionmentioning

confidence: 99%

Hypergeometric structures in Feynman integrals

Blümlein

Saragnese

Schneider

2023

Ann Math Artif Intell

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For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

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“…The parameterized version is used also in holonomic summation [6] and generalizations of it [5]. Further details can found, e.g., in [26].…”

Section: Introductionmentioning

confidence: 99%

Solving linear difference equations with coefficients in rings with idempotent representations

Ablinger¹,

Schneider²

2021

Preprint

Self Cite

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We introduce a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposed by a direct sum of integral domains (using idempotent elements) that enjoys certain technical features and that the coefficients of the difference equation are not degenerated. Using this mechanism we can reduce the problem to find solutions in a ring (with zerodivisors) to search solutions in several copies of integral domains. Utilizing existing solvers in this integral domain setting, we obtain a general solver where the components of the linear difference equations and the solutions can be taken from difference rings that are built e.g., by ΠΣ-extensions over ΠΣ-fields. This class of difference rings contains, e.g., nested sums and products, products over roots of unity and nested sums defined over such objects.

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Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

Cited by 4 publications

References 116 publications

Hypergeometric Structures in Feynman Integrals

Hypergeometric Structures in Feynman Integrals

Hypergeometric structures in Feynman integrals

Solving linear difference equations with coefficients in rings with idempotent representations

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