On an identity by Chaundy and Bullard. I

Abstract

An identity by Chaundy and Bullard writes 1/(1 − x) n ( n = l, 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 Gauβ 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 PDEs. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral.