Structural theorems for symbolic summation

Schneider, Carsten

doi:10.1007/s00200-009-0115-3

Cited by 34 publications

(44 citation statements)

References 52 publications

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“…In [53,59] depth-optimal ΠΣ * -extension have been introduced which refine the notion of reduced ΠΣ * -extensions and which give even stronger structural results than Theorem 8; see [62]. In particular one can search for such an improved ΠΣ * -field that leads to sum representations with minimal nesting depth [60].…”

Section: Resultsmentioning

confidence: 99%

“…Applying this transformation iteratively (see [62,Algorithm 1]) enables one to transform any ΠΣ * -extension to a reduced version.…”

Section: A Constructive Version Of Karr's Structural Theoremmentioning

confidence: 99%

“…7). The presented algorithms can be used to transform any ΠΣ * -field to a reduced version [62]. As a consequence, we obtain a constructive version of Karr's structural theorem, which can be considered as the discrete analogue of Liouville's Theorem [34] for indefinite integration.…”

Section: Introductionmentioning

confidence: 99%

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Computer Algebra and Polynomials

2015

Lecture Notes in Computer Science

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Section: Resultsmentioning

confidence: 99%

“…Applying this transformation iteratively (see [62,Algorithm 1]) enables one to transform any ΠΣ * -extension to a reduced version.…”

Section: A Constructive Version Of Karr's Structural Theoremmentioning

confidence: 99%

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Computer Algebra and Polynomials

2015

Lecture Notes in Computer Science

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“…A much more general algorithm which finds solutions in ΠΣ * -difference extension fields of (K(n), σ) is presented in [32]. For the relevant theory, see [33], [35], [36], [37].…”

Section: Letmentioning

confidence: 99%

Solving Linear Recurrence Equations with Polynomial Coefficients

Petkovšek

Zakrajšek

2013

Texts &Amp; Monographs in Symbolic Computation

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Summation is closely related to solving linear recurrence equations, since an indefinite sum satisfies a first-order linear recurrence with constant coefficients, and a definite proper-hypergeometric sum satisfies a linear recurrence with polynomial coefficients. Conversely, d'Alembertian solutions of linear recurrences can be expressed as nested indefinite sums with hypergeometric summands. We sketch the simplest algorithms for finding polynomial, rational, hypergeometric, d'Alembertian, and Liouvillian solutions of linear recurrences with polynomial coefficients, and refer to the relevant literature for state-of-the-art algorithms for these tasks. We outline an algorithm for finding the minimal annihilator of a given P-recursive sequence, prove the salient closure properties of d'Alembertian sequences, and present an alternative proof of a recent result of Reutenauer's that Liouvillian sequences are precisely the interlacings of d'Alembertian ones.

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“…As a side product, one can simplify the sum expressions w.r.t. certain optimality criteria, like finding sum representations with optimal nesting depth [35,38,40], with a minimal number of summation objects in the summands [37], or with minimal degrees arising in the numerators and denominators [33]. In particular, the occurring sums and products in the reduced expression are algebraically independent among each other [36,17,42].…”

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}.

show abstract

Structural theorems for symbolic summation

Cited by 34 publications

References 52 publications

Computer Algebra and Polynomials

Computer Algebra and Polynomials

Solving Linear Recurrence Equations with Polynomial Coefficients

Minimal Representations and Algebraic Relations for Single Nested Products

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