On an identity by Chaundy and Bullard. I

TL;DR

This paper surveys various proofs of a binomial series identity by Chaundy and Bullard, explores its extensions to complex parameters and multivariable cases, and relates it to hypergeometric functions and PDEs.

Contribution

It provides a comprehensive survey of proofs, discusses extensions to complex and multivariable cases, and introduces a new proof using Dirichlet's beta integral.

Findings

Connections with Gauss hypergeometric series clarified

Extension to complex parameters analyzed

Multivariable analogue proved using beta integral

Abstract

An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauss hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDE's. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral.