A Unified Reduction for Hypergeometric and q-Hypergeometric Creative Telescoping

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.