# On an identity by Chaundy and Bullard. I

### (20 pages)

**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.

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