On an identity by Chaundy and Bullard - different proofs and generalizations

On an identity by Chaundy and Bullard - different proofs and generalizations

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, a notable special case by Ingrid Daubechies, while she was setting up the theory of wavelets of  compact support. We discuss a number of different proofs of the identity, and explain its relationship with Gauß hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. For a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr, we provide a new proof by splitting up Dirichlet's multivariable beta integral. Finally, we present several generalizations, including q- and elliptic extensions, of the Chaundy-Bullard identity.

This is joint work with Tom Koornwinder.