# On an identity by Chaundy and Bullard. III. Basic and elliptic extensions

### (22 pages)

**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 Bézout 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.

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