On the existence of telescopers for mixed hypergeometric terms

Chen, Shaoshi; Chyzak, Frédéric; Feng, Ruyong; Fu, Guofeng; Li, Ziming

doi:10.1016/j.jsc.2014.08.005

Cited by 27 publications

(18 citation statements)

References 36 publications

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“…Lemma 5. 16. Let (F, σ) be a difference field with K = const(F, σ), and let (E, σ) be a basic RΠΣ * -extension of (F, σ) given in the form (4.2) where the R-monomial y has order n with α = σ(y) y ∈ K. Let e 0 , .…”

Section: Further Characterizations Of Rπς * -Extensions: the Interlacmentioning

confidence: 99%

Summation Theory II: Characterizations of RΠΣ⁎-extensions and algorithmic aspects

Schneider

2017

Journal of Symbolic Computation

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Recently, RΠΣ * -extensions have been introduced which extend Karr's ΠΣ * -fields substantially: one can represent expressions not only in terms of transcendental sums and products, but one can work also with products over primitive roots of unity. Since one can solve the parameterized telescoping problem in such rings, covering as special cases the summation paradigms of telescoping and creative telescoping, one obtains a rather flexible toolbox for symbolic summation. This article is the continuation of this work. Inspired by Singer's Galois theory of difference equations we will work out several alternative characterizations of RΠΣ * -extensions: adjoining naively sums and products leads to an RΠΣ * -extension iff the obtained difference ring is simple iff the ring can be embedded into the ring of sequences iff the ring can be given by the interlacing of ΠΣ * -extensions. From the viewpoint of applications this leads to a fully automatic machinery to represent indefinite nested sums and products in such RΠΣ * -rings. In addition, we work out how the parameterized telescoping paradigm can be used to prove algebraic independence of indefinite nested sums. Furthermore, one obtains an alternative reduction tactic to solve the parameterized telescoping problem in basic RΠΣ * -extensions exploiting the interlacing property.Key words: difference ring extensions, roots of unity, indefinite nested sums and products, simple difference rings, difference ideals, constants, interlacing of difference rings, embedding into the difference ring of sequences, Galois theory ⋆

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Section: Further Characterizations Of Rπς * -Extensions: the Interlacmentioning

confidence: 99%

Summation Theory II: Characterizations of RΠΣ⁎-extensions and algorithmic aspects

Schneider

2017

Journal of Symbolic Computation

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“…We now consider the exactness testing problem in the case when ∂ x = ∆ x,q and ∂ y = ∆ y . To this end, we first recall a lemma which is a special case of Lemma 5.4 in [10]. Let f ∈ k(x, y).…”

Section: Exactness Criteriamentioning

confidence: 99%

Bivariate Extensions of Abramov’s Algorithm for Rational Summation

Chen

2018

Advances in Computer Algebra

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Abramov's algorithm enables us to decide whether a univariate rational function can be written as a difference of another rational function, which has been a fundamental algorithm for rational summation. In 2014, Chen and Singer generalized Abramov's algorithm to the case of rational functions in two (q-)discrete variables. In this paper we solve the remaining three mixed cases, which completes our recent project on bivariate extensions of Abramov's algorithm for rational summation.

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“…Both articles also discuss the trading of order for degree, i.e., the option of computing an equation with lower coefficient degree at the cost of a larger order and vice versa; this trade-off can be used to reduce the complexity of the algorithms. The question of existence criteria for creative telescoping relations for mixed hypergeometric terms was answered in [26]. Concerning new creative telescoping algorithms, the use of residues for the computation of telescopers has been investigated in [30] for rational functions and in [29] for algebraic functions.…”

Section: History and Developmentsmentioning

confidence: 99%

Computer Algebra in Quantum Field Theory

Schneider¹,

Blümlein²

2013

Texts &Amp; Monographs in Symbolic Computation

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The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts.

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On the existence of telescopers for mixed hypergeometric terms

Abstract: We present a criterion for the existence of telescopers for mixed hypergeometric terms, which is based on multiplicative and additive decompositions. The criterion enables us to determine the termination of Zeilberger's algorithms for mixed hypergeometric inputs.

Cited by 27 publications

References 36 publications

Summation Theory II: Characterizations of RΠΣ⁎-extensions and algorithmic aspects

Summation Theory II: Characterizations of RΠΣ⁎-extensions and algorithmic aspects

Bivariate Extensions of Abramov’s Algorithm for Rational Summation

Computer Algebra in Quantum Field Theory

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