2023/04/19 by Natsuko Hoshi, Hoshi, Natsuko, Makoto Katori +5
Mathematics · #05A19 (Primary) 05A10 #05A30 #05C22 #05C81 #11B65 #33D15 #33E05 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2304.10003
openalex publication_date 2023/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.