A Unified Reduction for Hypergeometric and $q$-Hypergeometric Creative Telescoping

We adapt the theory of normal and special polynomials from symbolic integration to the summation setting and then build 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 us 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.