Combinatorial identities and hypergeometric series

TL;DR

This paper presents a method linking combinatorial identities to hypergeometric series, enabling proofs of identities by reducing complex series to simpler forms, with examples illustrating its effectiveness.

Contribution

It introduces a novel approach to connect combinatorial identities with hypergeometric series, facilitating their proof without relying on existing hypergeometric function tables.

Findings

Successfully proved several combinatorial identities using the hypergeometric series method

Reduced complex hypergeometric series to simpler forms in multiple cases

Demonstrated the method's potential to simplify proofs of combinatorial identities

Abstract

This paper describes a method to find a connection between combinatorial identities and hypergeometric series with a number of examples. Combinatorial identities can often be written as hypergeometric series with unit argument. In a number of cases these hypergeometric series are balanced and can be reduced to a simpler form. In this paper some combinatorial identities are proved using this method assuming that the results in the tables of Prudnikov et al. [12] are proven without using hypergeometric functions.