# Creative Telescoping for Holonomic Functions

@article{Koutschan2013CreativeTF, title={Creative Telescoping for Holonomic Functions}, author={Christoph Koutschan}, journal={ArXiv}, year={2013}, volume={abs/1307.4554} }

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…

## 46 Citations

Efficient Algorithms for Mixed Creative Telscoping

- MathematicsISSAC
- 2016

This work designs a new creative telescoping algorithm operating on this class of inputs, based on a Hermite-like reduction procedure, and focuses on the integration of bivariate hypergeometric-hyperexponential terms.

A Reduction Approach to Creative Telescoping

- MathematicsISSAC
- 2019

This tutorial will overview several reduction algorithms in symbolic integration and summation, explain the idea of creative telescoping via reductions, and present intriguing applications of this new approach.

Contiguous Relations and Creative Telescoping

- MathematicsTexts & Monographs in Symbolic Computation
- 2021

This article presents an algorithmic theory of contiguous relations. Contiguous relations, first studied by Gauß, are a fundamental concept within the theory of hypergeometric series. In contrast to…

The ABC of Creative Telescoping - Algorithms, Bounds, Complexity

- Computer Science
- 2014

In the present memoir, the evolution of algorithms and of the contributions in adapting the approach to larger and larger classes of functions are recounted, from the initial framework of hypergeometric sequences, given by first-order recurrences, to the case of functions given by higher-order equations, then to functionsgiven by positive-dimensional ideals.

Parallel telescoping and parameterized Picard-Vessiot theory

- MathematicsISSAC
- 2014

A necessary and sufficient condition guaranteeing the existence of parallel telescopers for differentially finite functions is presented, and an algorithm to compute minimal ones for compatible hyperexponential functions is developed.

Lazy Hermite Reduction and Creative Telescoping for Algebraic Functions

- MathematicsISSAC
- 2021

The lazy Hermite reduction is sharpen by combining it with the polynomial reduction to solve the decomposition problem of algebraic functions.

Convolution surfaces with varying radius: Formulae for skeletons made of arcs of circles and line segments

- Mathematics
- 2018

In skeleton-based geometric modeling, convolution is an established technique: smooth surfaces around a skeleton made of curves are given as the level set of a convolution field. Varying the radius…

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

- MathematicsJ. Symb. Comput.
- 2017

Binomial Determinants for Tiling Problems Yield to the Holonomic Ansatz

- MathematicsEur. J. Comb.
- 2022

## References

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This note reinvestigate the task of computing creative telescoping relations in differential–difference operator algebras using an ansatz that explicitly includes the denominators of the delta parts and shows that it can be superior to existing methods by a large factor.

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This work describes a precise and elementary algorithmic version of the Griffiths-Dwork method for the creative telescoping of rational functions that leads to bounds on the order and degree of the coefficients of the differential equation, and to the first complexity result which is single exponential in the number of variables.

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This work blends the general method of creative telescoping with the well-known Hermite reduction, and uses this blended method to compute diagonals of rational power series arising from combinatorics.

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The authors' bounds are expressed as curves in the (r, d)-plane which assign to every order r a bound on the degree d of the telescopers, which reflect the phenomenon that higher order telescopers tend to have lower degree, and vice versa.

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This thesis contributed to translating Takayama's algorithm into a new context, in order to apply it to an until then open problem, the proof of Ira Gessel's lattice path conjecture, and to make the underlying computations feasible the authors employed a new approach for finding creative telescoping operators.

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The method of creative telescoping is applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma function or polylogarithms, which are not covered by the holonomic framework.