On an identity of Chaundy and Bullard. III. Basic and elliptic extensions

Abstract

The identity by Chaundy and Bullard expresses \(1\) as a sum of two truncated binomial series in one variable where the truncations depend on two different non-negative integers. We present basic and elliptic extensions of the Chaundy--Bullard identity. The most general result, the elliptic extension, involves, in addition to the nome \(p\) and the base \(q\), four independent complex variables. Our proof uses a suitable weighted lattice path model. We also show how three of the basic extensions can be viewed as B\'ezout identities. Inspired by the lattice path model, we give a new elliptic extension of the binomial theorem, taking the form of an identity for elliptic commuting variables. We further present variants of the homogeneous form of the identity for \(q\)-commuting and for elliptic commuting variables.