A Generalization of the Chu-Vandermonde Convolution and some Harmonic Number Identities

TL;DR

This paper generalizes the Chu-Vandermonde convolution, introduces related identities, and derives harmonic number identities, providing new tools and recursive formulas for combinatorial sums.

Contribution

It presents a new generalization of the Chu-Vandermonde convolution, along with related identities and harmonic number identities, expanding the theoretical framework.

Findings

Derived a generalized Chu-Vandermonde convolution identity.

Established related identities and harmonic number identities.

Developed a recursion formula for a combinatorial sum.

Abstract

A generalization of the Chu-Vandermonde convolution is presented and proved with the integral representation method. This identity can be transformed into another identity, which has as special cases two known identities. Another identity that is closely related to this identity is presented and proved. Using the modified geometric series from another paper, some closely related identities are listed. Some corresponding harmonic number identities are derived, which have as special cases some known harmonic number identities. For one combinatorial sum a recursion formula is derived and used to compute a few examples.