Multiple Rogers-Ramanujan type identities for inert quadratic orders
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911261978001408 |
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| author | Chern, Shane Huang, Yifeng |
| author_facet | Chern, Shane Huang, Yifeng |
| contents | We compute the Quot and finitized Coh zeta functions of the inert quadratic orders $\mathbb{F}_q[[T]]+T^{m}\mathbb{F}_{q^{2}}[[T]]$ for every $m\geq 1$ in terms of a $2m$-fold multisum, and then show this multisum equals an $m$-fold Bressoud sum. This proves a recent conjecture of the second author, rounding up the line of exploration in the series of work by the authors and Jiang. The equality between the $2m$-fold multisum and the $m$-fold Bressoud sum is built upon generalizing the multisum by introducing a ``ghost'' parameter $a$ to its summands. We then show that such an $a$-generalization is surprisingly $a$-independent by purely $q$-theoretic techniques. Finally, we propose a refined multisum that interpolates two versions of Quot zeta functions for all three types of quadratic orders. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple Rogers-Ramanujan type identities for inert quadratic orders Chern, Shane Huang, Yifeng Algebraic Geometry Combinatorics Number Theory 14D20, 11S45, 11P84, 05A15, 33D15 We compute the Quot and finitized Coh zeta functions of the inert quadratic orders $\mathbb{F}_q[[T]]+T^{m}\mathbb{F}_{q^{2}}[[T]]$ for every $m\geq 1$ in terms of a $2m$-fold multisum, and then show this multisum equals an $m$-fold Bressoud sum. This proves a recent conjecture of the second author, rounding up the line of exploration in the series of work by the authors and Jiang. The equality between the $2m$-fold multisum and the $m$-fold Bressoud sum is built upon generalizing the multisum by introducing a ``ghost'' parameter $a$ to its summands. We then show that such an $a$-generalization is surprisingly $a$-independent by purely $q$-theoretic techniques. Finally, we propose a refined multisum that interpolates two versions of Quot zeta functions for all three types of quadratic orders. |
| title | Multiple Rogers-Ramanujan type identities for inert quadratic orders |
| topic | Algebraic Geometry Combinatorics Number Theory 14D20, 11S45, 11P84, 05A15, 33D15 |
| url | https://arxiv.org/abs/2511.09452 |