Identities on Zagier's rank two examples for Nahm's problem
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910434184921088 |
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| author | Wang, Liuquan |
| author_facet | Wang, Liuquan |
| contents | Let $r\geq 1$ be a positive integer, $A$ a real positive definite symmetric $r\times r$ matrix, $B$ a vector of length $r$, and $C$ a scalar. Nahm's problem is to describe all such $A,B$ and $C$ with rational entries for which a specific $r$-fold $q$-hypergeometric series (denoted by $f_{A,B,C}(q)$) involving the parameters $A,B,C$ is modular. When the rank $r=2$, Zagier provided eleven sets of examples of $(A,B,C)$ for which $f_{A,B,C}(q)$ is likely to be modular. We present a number of Rogers--Ramanujan type identities involving double sums, which give modular representations for Zagier's rank two examples. Together with several known cases in the literature, we verified ten of Zagier's examples and give conjectural identities for the remaining example. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_10748 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Identities on Zagier's rank two examples for Nahm's problem Wang, Liuquan Number Theory Classical Analysis and ODEs Combinatorics 11P84, 33D15, 33D60, 11F03 Let $r\geq 1$ be a positive integer, $A$ a real positive definite symmetric $r\times r$ matrix, $B$ a vector of length $r$, and $C$ a scalar. Nahm's problem is to describe all such $A,B$ and $C$ with rational entries for which a specific $r$-fold $q$-hypergeometric series (denoted by $f_{A,B,C}(q)$) involving the parameters $A,B,C$ is modular. When the rank $r=2$, Zagier provided eleven sets of examples of $(A,B,C)$ for which $f_{A,B,C}(q)$ is likely to be modular. We present a number of Rogers--Ramanujan type identities involving double sums, which give modular representations for Zagier's rank two examples. Together with several known cases in the literature, we verified ten of Zagier's examples and give conjectural identities for the remaining example. |
| title | Identities on Zagier's rank two examples for Nahm's problem |
| topic | Number Theory Classical Analysis and ODEs Combinatorics 11P84, 33D15, 33D60, 11F03 |
| url | https://arxiv.org/abs/2210.10748 |