Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli
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| Format: | Preprint |
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2025
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| _version_ | 1866909854880235520 |
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| author | Alamoudi, Yazan Alladi, Krishnaswami |
| author_facet | Alamoudi, Yazan Alladi, Krishnaswami |
| contents | Here, we establish a polynomial identity in three variables $a, b, c$, and with the degree of the polynomial given in terms of two integers $L, M$. By letting $L$ and $M$ tend to infinity, we get the 1993 Alladi-Gordon $q$-hypergeometric key-identity for the generalized Schur Theorem as well as the fundamental Lebesgue identity by two different choices of the variables. This polynomial identity provides a generalization and a unified approach to the Schur and Lebesgue theorems. We discuss other analytic identities for the Lebesgue and Schur theorems and also provide a key identity ($q$-hypergeometric) for Andrews' deep refinement of the Alladi-Schur theorem. Finally, we discuss a new infinite hierarchy of identities, the first three of which relate to the partition theorems of Euler, Lebesgue, and Capparelli, and provide their polynomial versions as well. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_04712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli Alamoudi, Yazan Alladi, Krishnaswami Number Theory Combinatorics 05A15, 05A17, 11P81, 11P83 Here, we establish a polynomial identity in three variables $a, b, c$, and with the degree of the polynomial given in terms of two integers $L, M$. By letting $L$ and $M$ tend to infinity, we get the 1993 Alladi-Gordon $q$-hypergeometric key-identity for the generalized Schur Theorem as well as the fundamental Lebesgue identity by two different choices of the variables. This polynomial identity provides a generalization and a unified approach to the Schur and Lebesgue theorems. We discuss other analytic identities for the Lebesgue and Schur theorems and also provide a key identity ($q$-hypergeometric) for Andrews' deep refinement of the Alladi-Schur theorem. Finally, we discuss a new infinite hierarchy of identities, the first three of which relate to the partition theorems of Euler, Lebesgue, and Capparelli, and provide their polynomial versions as well. |
| title | Some $q$-hypergeometric identities associated with partition theorems of Lebesgue, Schur and Capparelli |
| topic | Number Theory Combinatorics 05A15, 05A17, 11P81, 11P83 |
| url | https://arxiv.org/abs/2502.04712 |