Natsuko Hoshi, Makato Katori, Tom. H. Koornwinder and Michael J. Schlosser

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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