Gosper Summability of Rational Multiples of Hypergeometric Terms

TL;DR

This paper explores the Gosper summability of rational multiples of hypergeometric terms, providing bounds, methods for construction, and applications to super-congruences and q-analogues, advancing the understanding of hypergeometric series summation.

Contribution

It introduces bounds on the degree of rational functions for Gosper summability and offers a systematic method to construct summable series from known ones.

Findings

Bounds on numerator degree for Gosper summability

Method to determine denominator from Gosper representation

Application to super-congruences and q-analogues

Abstract

By telescoping method, Sun gave some hypergeometric series whose sums are related to $\pi$ recently. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term $t_k$, we consider the Gosper summability of $r(k)t_k$ for $r(k)$ being a rational function of $k$. We give an upper bound and a lower bound on the degree of the numerator of $r(k)$ such that $r(k)t_k$ is Gosper summable. We also show that the denominator of the $r(k)$ can read from the Gosper representation of $t_{k+1}/t_k$. Based on these results, we give a systematic method to construct series whose sums can be derived from the known ones. We also illustrated the corresponding super-congruences and the $q$-analogue of the approach.