Algebraic and algorithmic aspects of linear
                    difference equations

Singer, Michael F.

doi:10.1090/surv/211/01

Cited by 5 publications

(2 citation statements)

References 19 publications

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“…Further simplification aspects within ΠΣ-fields (e.g., finding products where the degrees of the top most sum or product in the numerator and denominator of a multiplicand are minimal) are elaborated in [25,103]. In addition, methods to find algebraic relations of sequences built by products are given in [73,84,108,119,123].…”

Section: +mentioning

confidence: 99%

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

Schneider¹

2021

Preprint

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A general overview of the existing difference ring theory for symbolic summation is given. Special emphasis is put on the user interface: the translation and back translation of the corresponding representations within the term algebra and the formal difference ring setting. In particular, canonical (unique) representations and their refinements in the introduced term algebra are explored by utilizing the available difference ring theory. Based on that, precise input-output specifications of the available tools of the summation package Sigma are provided. * This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 764850, SAGEX and from the Austrian Science Foundation (FWF) grant SFB F50 (F5005-N15) in the framework of the Special Research Program "Algorithmic and Enumerative Combinatorics".( ) ∈ Ã( ) and to find all rational solutions of Ã( ) of given linear recurrence (4) with ( ), ( ) ∈ Ã( ). In particular, Gosper's telescoping algorithm [61] for hypergeometric products ( ) = = ( ) with ( ) ∈ Ã( ) and Zeilberger's extension to definite sums via his creative telescoping paradigm [52,53,85,89,91,131]

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Section: +mentioning

confidence: 99%

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

Schneider¹

2021

Preprint

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“…This result is connected to [20,43] where all relations of a finite set of sequences can be computed that satisfy homogeneous linear recurrences with constant coefficients. In particular in [20] their algorithm is reduced to find all relations of a finite set of geometric sequences, i.e., sequences produced by the products in (1) with f 1 (k), .…”

Section: Introductionmentioning

confidence: 99%

Minimal Representations and Algebraic Relations for Single Nested Products

Schneider

2020

Program Comput Soft

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Recently, it has been shown constructively how a finite set of hypergeometric products, multibasic hypergeometric products or their mixed versions can be modeled properly in the setting of formal difference rings. Here special emphasis is put on robust constructions: whenever further products have to be considered, one can reuse -up to some mild modifications-the already existing difference ring. In this article we relax this robustness criteria and seek for another form of optimality. We will elaborate a general framework to represent a finite set of products in a formal difference ring where the number of transcendental product generators is minimal. As a bonus we are able to describe explicitly all relations among the given input products.This work was supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15).3 In this example the evaluation of an element from É(ι)(x) is carried out by replacing x with concrete values n ∈ AE. Later we will generalize this simplest case to formal difference rings equipped with an evaluation function acting on the ring elements.4 More generally, if R is a commutative ring with 1, we define the ideal I generated by a 1 , . . . , ar ∈ R with I = a 1 , . . . , ar R = {f 1 a 1 + · · · + fr ar | f 1 , . . . , fr ∈ R}.

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

Schneider

2021

Texts &Amp; Monographs in Symbolic Computation

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Algebraic and algorithmic aspects of linear difference equations

Abstract: These are an extended version of lecture notes from a series of talks given at the

Cited by 5 publications

References 19 publications

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

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

Minimal Representations and Algebraic Relations for Single Nested Products

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

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