A Unified Reduction for Hypergeometric and q-Hypergeometric Creative Telescoping

TL;DR

This paper introduces a unified reduction and creative telescoping algorithm that applies to both hypergeometric and q-hypergeometric terms, highlighting their differences and enabling efficient symbolic summation.

Contribution

It develops a general framework and algorithms that unify the treatment of hypergeometric and q-hypergeometric summation problems, revealing their intrinsic differences.

Findings

Algorithms work for both hypergeometric and q-hypergeometric cases

Efficient splitting of shift and q-shift cases when necessary

Computational experiments demonstrate effectiveness

Abstract

We adapt the theory of normal and special polynomials from symbolic integration to the summation setting, and then built up a general framework embracing both the usual shift case and the $q$-shift case. In the context of this general framework, we develop a unified reduction algorithm, and subsequently a creative telescoping algorithm, applicable to both hypergeometric terms and their $q$-analogues. Our algorithms allow to split up the usual shift case and the $q$-shift case only when it is really necessary, and thus instantly reveal the intrinsic differences between these two cases. Computational experiments are also provided.