On an identity of Chaundy and Bullard. III. Basic and elliptic extensions
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866918096584835072 |
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| author | Hoshi, Natsuko Katori, Makoto Koornwinder, Tom H. Schlosser, Michael J. |
| author_facet | Hoshi, Natsuko Katori, Makoto Koornwinder, Tom H. Schlosser, Michael J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10003 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On an identity of Chaundy and Bullard. III. Basic and elliptic extensions Hoshi, Natsuko Katori, Makoto Koornwinder, Tom H. Schlosser, Michael J. Combinatorics Quantum Algebra 05A19 (Primary) 05A10, 05A30, 05C22, 05C81, 11B65, 33D15, 33E05 (Secondary) 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. |
| title | On an identity of Chaundy and Bullard. III. Basic and elliptic extensions |
| topic | Combinatorics Quantum Algebra 05A19 (Primary) 05A10, 05A30, 05C22, 05C81, 11B65, 33D15, 33E05 (Secondary) |
| url | https://arxiv.org/abs/2304.10003 |