Some Nice Sums are Almost as Nice if you turn them Upside Down

TL;DR

This paper explores various binomial and q-hypergeometric sums, expressing them as products of indefinite hypergeometric sums and closed-form hypergeometric sequences, revealing new structural insights.

Contribution

It introduces a novel representation of complex sums as products of indefinite hypergeometric sums and hypergeometric sequences, advancing understanding of their structure.

Findings

New representations of binomial and q-hypergeometric sums

Connections between sums and hypergeometric functions

Enhanced understanding of Dixon's identity sums

Abstract

We represent the sums $\sum_{k=0}^{n-1}{n \choose k}^{-2}$, $\sum_{k=0}^m{m\choose k}^{-1}{a\choose n-k}^{-1}$, $\sum_{k=0}^{n-1}\frac{q^{-k(k-1)}}{{\genfrac{[}{]}{0pt}{}{n}{k}}_q}$, and the sum of the reciprocals of the summands in Dixon's identity, each as a product of an {\it indefinite hypergeometric sum} times a (closed form) {\it hypergeometric sequence}