Hermite reduction and creative telescoping for hyperexponential functions

The Apagodu-Zeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary ∂-finite functions. In analogy to the hypergeometric case, we introduce the notion of proper ∂-finite functions. We show that the algorithm always succeeds for these functions, and we give a tight a priori bound for the order of the output operator.

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“…. , 5) and linear combinations f = p e 1 + · · · + p en with n rational exponents with pairwise coprime denominators (n = 1, . .…”

Section: Differential Casementioning

confidence: 99%

A generalized Apagodu-Zeilberger algorithm

Chen

Kauers²,

Koutschan

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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“…Our generalized Hermite reduction is inspired by these works. It has the same architecture as several previous ones [10,16,11,44]: local reductions at finite places, followed by a reduction at infinity and the computation of an exceptional set to obtain a canonical form. Our first contribution in the present paper is to open a new direction of generalization, namely by considering reductions with respect to other operators in K(x) ∂ x than the derivation operator ∂ x , acting on the space K(x) of rational functions.…”

Section: Previous Workmentioning

confidence: 99%

Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions

Bostan

Chyzak

Lairez

et al. 2018

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

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Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. We extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping.

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“…A second, alternate way to ensure termination, used for example in [11,12], is to show that, for a given function f , the remainders red(f ), red(σ x (f )), red(σ 2…”

Section: Introductionmentioning

confidence: 99%

Existence Problem of Telescopers for Rational Functions in Three Variables

Chen

Zhu

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach.

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Hermite reduction and creative telescoping for hyperexponential functions

Cited by 53 publications

References 16 publications

A generalized Apagodu-Zeilberger algorithm

A generalized Apagodu-Zeilberger algorithm

Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions

Existence Problem of Telescopers for Rational Functions in Three Variables

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