On a conjecture due to Kanade related to Nahm sums
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917315071705088 |
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| author | Hakimoglu-Brown, Cetin |
| author_facet | Hakimoglu-Brown, Cetin |
| contents | Kanade explored the construction of modular companions to $q$-series identities, using the asymptotics of Nahm sums, and Mizuno
[Ramanujan J.\ {\bf 66} (2025), Paper No.\ 62, 31] recently obtained a generalization of Kanade's asymptotic formula for symmetrizable
Nahm sums. A related conjecture from Kanade concerning the dilogarithm function and related to the work of Kur\c sungöz on
Andrews--{G}ordon-type series [Ann.\ Comb.\ {\bf 23} (2019), 835--888] has remained open. In this paper, we prove Kanade's
conjecture, through an application of dilogarithm identities due to Kirillov together with a dilogarithm ladder due to Lewin
and Loxton. Inspired by Kanade's result, we extend this to conjecture two new dilogarithm identities and associated rank-2 matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00836 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a conjecture due to Kanade related to Nahm sums Hakimoglu-Brown, Cetin Number Theory 33B30, 33B15 Kanade explored the construction of modular companions to $q$-series identities, using the asymptotics of Nahm sums, and Mizuno [Ramanujan J.\ {\bf 66} (2025), Paper No.\ 62, 31] recently obtained a generalization of Kanade's asymptotic formula for symmetrizable Nahm sums. A related conjecture from Kanade concerning the dilogarithm function and related to the work of Kur\c sungöz on Andrews--{G}ordon-type series [Ann.\ Comb.\ {\bf 23} (2019), 835--888] has remained open. In this paper, we prove Kanade's conjecture, through an application of dilogarithm identities due to Kirillov together with a dilogarithm ladder due to Lewin and Loxton. Inspired by Kanade's result, we extend this to conjecture two new dilogarithm identities and associated rank-2 matrices. |
| title | On a conjecture due to Kanade related to Nahm sums |
| topic | Number Theory 33B30, 33B15 |
| url | https://arxiv.org/abs/2603.00836 |