Identities on Zagier’s rank two examples for Nahm’s problem.(English)Zbl 07875607
Summary: 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.
MSC:
| 11P84 | Partition identities; identities of Rogers-Ramanujan type |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
| 33D60 | Basic hypergeometric integrals and functions defined by them |
| 05A19 | Combinatorial identities, bijective combinatorics |
Keywords:
Nahm’s problem;Rogers-Ramanujan type identities;sum-product identities;integral method;Slater’s listCite
References:
| [1] | Andrews, GE, On the General Rogers-Ramanujan Theorem, 1974, Providence, RI: Amer. Math. Soc, Providence, RI ·Zbl 0296.10010 ·doi:10.1090/memo/0152 |
| [2] | Andrews, G.E.: The Theory of Partitions. Addison-Wesley (1976). Reissued Cambridge (1998) ·Zbl 0371.10001 |
| [3] | Andrews, GE; Berndt, BC, Ramanujan’s Lost Notebook, Part II, 2009, Berlin: Springer, Berlin ·Zbl 1180.11001 |
| [4] | Andrews, G.E., Uncu, A.K.: Sequences in overpartitions. Ramanujan J. 61, 715-729 (2023) ·Zbl 1547.11117 |
| [5] | Calinescu, C.; Milas, A.; Penn, M., Vertex algebraic structure of principal subspaces of basic \(A_{2n}^{(2)}\)-modules, J. Pure Appl. Algebra, 220, 1752-1784, 2016 ·Zbl 1395.17059 ·doi:10.1016/j.jpaa.2015.10.001 |
| [6] | Cherednik, I.; Feigin, B., Rogers-Ramanujan type identities and Nil-DAHA, Adv. Math., 248, 1050-1088, 2013 ·Zbl 1298.33029 ·doi:10.1016/j.aim.2013.08.025 |
| [7] | Chern, S., Asymmetric Rogers-Ramanujan type identities. I, The Andrews-Uncu Conjecture, Proc. Am. Math. Soc., 151, 3269-3279, 2023 ·Zbl 1523.11188 |
| [8] | Cao, Z.; Wang, L., Multi-sum Rogers-Ramanujan type identities, J. Math. Anal. Appl., 522, 2, 2023 ·Zbl 1516.05011 ·doi:10.1016/j.jmaa.2022.126960 |
| [9] | Cao, Z.; Rosengren, H.; Wang, L., On some double Nahm sums of Zagier, J. Combin. Theory Ser. A, 202, 2024 ·Zbl 1531.05016 ·doi:10.1016/j.jcta.2023.105819 |
| [10] | Frye, J., Garvan, F.G.: Automatic proof of theta-function identities, Elliptic integrals, elliptic functions and modular forms in quantum field theory. In: Texts Monogr. Symbol. Comput. (pp. 195-258). Springer, Cham (2019) |
| [11] | Gasper, G.; Rahman, M., Basic hypergeometric series, Encyclopedia of Mathematics and Its Applications, 2004, Cambridge: Cambridge University Press, Cambridge ·Zbl 1129.33005 |
| [12] | Gordon, B., A combinatorial generalization of the Rogers-Ramanujan identities, Am. J. Math., 83, 393-399, 1961 ·Zbl 0100.27303 ·doi:10.2307/2372962 |
| [13] | Kanade, S.; Russell, MC, Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type, Electron. J. Combin., 26, 1-6, 2019 ·Zbl 1409.05018 ·doi:10.37236/7847 |
| [14] | Lee, C.-H.: Algebraic structures in modular \(q\)-hypergeometric series. PhD Thesis, University of California, Berkeley (2012) |
| [15] | Mc Laughlin, J., Some more identities of Kanade-Russell type derived using Rosengren’s method, Ann. Comb., 27, 329-352, 2023 ·Zbl 1530.33017 ·doi:10.1007/s00026-022-00586-3 |
| [16] | Nahm, W.: Conformal field theory and the dilogarithm. In: 11th International Conference on Mathematical Physics (ICMP-11) (Satelite colloquia: New Problems in General Theory of Fields and Particles), Paris, pp. 662-667 (1994) ·Zbl 1052.81611 |
| [17] | Nahm, W.: Conformal field theory, dilogarithms and three dimensional manifold (Proceedings, Conference in Hangzhou, People’s Republic of China, September 1993). In: Nahm, W., Shen, J.-M. (eds.) Interface between Physics and Mathematics, pp. 154-165. World Scientific, Singapore (1994) |
| [18] | Nahm, W.: Conformal field theory and torsion elements of the Bloch group. In: Frontiers in Number Theory, Physics and Geometry, II, pp. 67-132. Springer (2007) ·Zbl 1193.81092 |
| [19] | Robins, S.: Generalized Dedekind \(\eta \)-products. In: The Rademacher Legacy to Mathematics (University Park, PA, 1992). Contemporary Mathematics, vol. 166, pp. 119-128. American Mathematical Society, Providence (1994) ·Zbl 0808.11031 |
| [20] | Rosengren, H., Proofs of some partition identities conjectured by Kanade and Russell, Ramanujan J., 61, 295-317, 2023 ·Zbl 1523.11189 ·doi:10.1007/s11139-021-00389-9 |
| [21] | Sills, AV, An Invitation to the Rogers-Ramanujan Identities, 2018, Cambridge: CRC Press, Cambridge ·Zbl 1429.11002 |
| [22] | Slater, LJ, Further identities of the Rogers-Ramanujan type, Proc. Lond. Math. Soc. (2), 54, 1, 147-167, 1952 ·Zbl 0046.27204 ·doi:10.1112/plms/s2-54.2.147 |
| [23] | Terhoeven, M.: Rationale konforme Feldtheorien, der Dilogarithmus und Invarianten von 3-Mannigfaltigkeiten, PhD Thesis Universitat Bonn (1995) |
| [24] | Uncu, AK, On double sum generating functions in connection with some classical partition theorems, Disc. Math., 344, 2021 ·Zbl 1472.05020 ·doi:10.1016/j.disc.2021.112562 |
| [25] | Vlasenko, M.; Zwegers, S., Nahm’s conjecture: asymptotic computations and counterexamples, Commun. Number Theory Phys., 5, 3, 617-642, 2011 ·Zbl 1256.81102 ·doi:10.4310/CNTP.2011.v5.n3.a2 |
| [26] | Wang, L., New proofs of some double sum Rogers-Ramanujan type identities, Ramanujan J., 62, 251-272, 2023 ·Zbl 07739356 ·doi:10.1007/s11139-022-00654-5 |
| [27] | Warnaar, SO, The generalized Borwein conjecture. II. Refined \(q\)-trinomial coefficients, Disc. Math., 272, 215-258, 2003 ·Zbl 1030.05004 ·doi:10.1016/S0012-365X(03)00047-5 |
| [28] | Zagier, D.: The dilogarithm function. In: Frontiers in Number Theory, Physics and Geometry, II, pp. 3-65. Springer (2007) ·Zbl 1176.11026 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
