A criterion for the applicability of Zeilberger's algorithm to rational functions

Абрамов, С. А.; Le, H. Q.

doi:10.1016/s0012-365x(02)00442-9

Cited by 37 publications

(25 citation statements)

References 14 publications

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“…According to the different choices of L and ∂ x , we have nine types of telescopers in general, see Table 1. The existence problem of telescopers is related to the termination of Zeilbergerstyle algorithms and has been studied in [8,6,17,15] but, to our knowledge, our results concerning telescopers of the six types underlined in the above table are new. In this section, we will present a unified way to solve this problem for rational functions by using the knowledge in the previous sections.…”

Section: Telescopersmentioning

confidence: 97%

“…However, the situation in other cases turns out to be more involved. For the rational function f = 1/(t 2 +x 2 ), Abramov and Le [37,8] showed that there is no telescoper in k(t) S t such that L(f ) = ∆ x (g) for any g ∈ k(t, x). In other cases, there are two main reasons for non-existence: one is the non-commutativity between linear operators ∂ t ∈ {D t , S t , Q t } and residue mappings, the other is that not all algebraic functions would satisfy linear (q)-recurrence relations.…”

Section: Existence Of Telescopersmentioning

confidence: 99%

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Residues and telescopers for bivariate rational functions

Chen

Singer

2012

Advances in Applied Mathematics

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We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discrete settings and characterize which operators can occur as telescopers. Using this latter characterization, we reprove results of Furstenberg and Zeilberger concerning diagonals of power series representing rational functions. The key concept behind these considerations is a generalization of the notion of residue in the continuous case to an analogous concept in the discrete and q-discrete cases.

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Section: Telescopersmentioning

confidence: 97%

Section: Existence Of Telescopersmentioning

confidence: 99%

Residues and telescopers for bivariate rational functions

Chen

Singer

2012

Advances in Applied Mathematics

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“…The notion creative telescoping was first coined by van der Poorten in his essay [92] on Apéry's proof of the irrationality of ζ (3). But certainly, the underlying principle was known and used long before as an ad hoc trick to tackle sums and integrals.…”

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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“…The termination problem of Zeilberger's algorithms has been extensively studied in the last two decades (Wilf and Zeilberger, 1992b;Abramov and Le, 2002;Abramov, 2003;Chen et al, 2005) and can be related to existence problems for other operations, like the computation of diagonals (Lipshitz, 1988). The main output of creative telescoping is the recurrence on the sum U .…”

Section: Introductionmentioning

confidence: 99%

“…However, the situation in other cases turns out to be more involved. In the discrete case, the first complete solution to the termination problem has been given by Le (2001) and Abramov and Le (2002), by deciding whether telescopers exist for a given bivariate rational sequence in the (q)-discrete variables y 1 and y 2 . According to their criterion, the rational sequence f = 1 y 2 1 + y 2 2 has no telescoper.…”

Section: Introductionmentioning

confidence: 99%