Explicit forms and proofs of Zagier’s rank three examples for Nahm’s problem.(English)Zbl 07872171
Recall the usual \(q\)-series notation, \[(a;q)_n = \prod_{j=0}^{n-1} (1-aq^j) \]and \[(a;q)_{\infty} = \prod_{j \geq 0} (1-aq^j). \]A general problem is to determine when the \(q\)-series \[f_{A,B,C}(q) = \sum_{n = (n_1,\dots , n_r)^T \in (\mathbb{Z}_{\geq 0})^r} \frac{q^{\frac{1}{2}n^TAn + Bn + C}}{(q;q)_{n_1} \cdots (q;q)_{n_r}} \]is a modular form. Here \(A\) is a matrix, \(B\) is a vector, and \(C\) is a scalar. Nahm’s problem refers to the case where \(A\) is positive definite.
When \(r=3\),D. Zagier [Frontiers in number theory, physics, and geometry II. On conformal field theories, discrete groups and renormalization. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer. 3–65 (2007;Zbl 1176.11026)] gave a list of twelve positive-definite or positive semi-definite matrices \(A\) along with several \(B\) and \(C\) for each matrix and conjectured that the corresponding series \(f_{A,B,C}(q)\) are modular. He proved his conjecture for three of the matrices \(A\). In this paper the author establishes the remaining cases. This amounts to proving explicit identities like \[\sum_{i,j,k \geq 0} \frac{q^{i^2+j^2+k^2 + ij+ik+i+j}}{(q;q)_i(q;q)_j(q;q)_k} = \frac{(q^4;q^4)_{\infty}^2}{(q;q)_{\infty}(q^2;q^2)_{\infty}}\]or \[\sum_{i,j,k \geq 0} \frac{q^{4i^2+2j^2+k^2 +4ij-2ik-2jk + 4i+2j}}{(q^2;q^2)_i(q^2;q^2)_j(q^2;q^2)_k} = \frac{(q^2;q^2)_{\infty}^3(q;q^{12})_{\infty}(q^{11};q^{12})_{\infty}(q^{12};q^{12})_{\infty}}{(q;q)_{\infty}^2(q^4;q^4)_{\infty}},\]where the modularity of the infinite products is well known. Most of the identities are proved by reducing the triple sum to a more tractable single or double sum.
When \(r=3\),D. Zagier [Frontiers in number theory, physics, and geometry II. On conformal field theories, discrete groups and renormalization. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer. 3–65 (2007;Zbl 1176.11026)] gave a list of twelve positive-definite or positive semi-definite matrices \(A\) along with several \(B\) and \(C\) for each matrix and conjectured that the corresponding series \(f_{A,B,C}(q)\) are modular. He proved his conjecture for three of the matrices \(A\). In this paper the author establishes the remaining cases. This amounts to proving explicit identities like \[\sum_{i,j,k \geq 0} \frac{q^{i^2+j^2+k^2 + ij+ik+i+j}}{(q;q)_i(q;q)_j(q;q)_k} = \frac{(q^4;q^4)_{\infty}^2}{(q;q)_{\infty}(q^2;q^2)_{\infty}}\]or \[\sum_{i,j,k \geq 0} \frac{q^{4i^2+2j^2+k^2 +4ij-2ik-2jk + 4i+2j}}{(q^2;q^2)_i(q^2;q^2)_j(q^2;q^2)_k} = \frac{(q^2;q^2)_{\infty}^3(q;q^{12})_{\infty}(q^{11};q^{12})_{\infty}(q^{12};q^{12})_{\infty}}{(q;q)_{\infty}^2(q^4;q^4)_{\infty}},\]where the modularity of the infinite products is well known. Most of the identities are proved by reducing the triple sum to a more tractable single or double sum.
Reviewer: Jeremy Lovejoy (Paris)
MSC:
| 11P84 | Partition identities; identities of Rogers-Ramanujan type |
| 11F03 | Modular and automorphic functions |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
| 33D60 | Basic hypergeometric integrals and functions defined by them |
Keywords:
Nahm’s conjecture;Rogers-Ramanujan type identities;sum-product identities;integral method;Slater’s list;Bressoud polynomialCitations:
Zbl 1176.11026Cite
References:
| [1] | Andrews, G. E., On the General Rogers-Ramanujan Theorem, 1974, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI ·Zbl 0296.10010 |
| [2] | Andrews, G. E., Problem 74-12, SIAM Rev., 16, 390, 1974 |
| [3] | Andrews, G. E., Multiple q-series identities, Houst. J. Math., 3, 1-13, 1981 |
| [4] | Andrews, G. E., q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, (CBMS Regional Conference Series in Mathematics, vol. 66, 1986, Amer. Math. Soc: Amer. Math. Soc Providence, RI) ·Zbl 0594.33001 |
| [5] | Andrews, G. E., Multiple series Rogers-Ramanujan type identities, Pac. J. Math., 114, 267-283, 1984 ·Zbl 0547.10012 |
| [6] | Andrews, G. E., The Theory of Partitions, 1998, Addison-Wesley: Reissued: Addison-Wesley: Reissued Cambridge, Reissued ·Zbl 0996.11002 |
| [7] | Andrews, G. E.; Berndt, B. C., Ramanujan’s Lost Notebook, Part II, 2009, Springer ·Zbl 1180.11001 |
| [8] | Andrews, G. E.; Schilling, A.; Warnaar, S. O., An \(A_2\) Bailey lemma and Rogers-Ramanujan-type identities, J. Am. Math. Soc., 12, 3, 677-702, 1999 ·Zbl 0917.05010 |
| [9] | Andrews, G. E.; Uncu, A. K., Sequences in overpartitions, Ramanujan J., 61, 715-729, 2023 ·Zbl 1547.11117 |
| [10] | Bowman, D.; Mc Laughlin, J.; Sills, A. V., Some more identities of Rogers-Ramanujan type, Ramanujan J., 18, 3, 307-325, 2009 ·Zbl 1196.11141 |
| [11] | Bressoud, D. M., A generalization of the Rogers-Ramanujan identities for all moduli, J. Comb. Theory, Ser. A, 27, 64-68, 1979 ·Zbl 0416.10009 |
| [12] | Bressoud, D. M., Some identities for terminating q-series, Math. Proc. Camb. Philos. Soc., 81, 211-223, 1981 ·Zbl 0454.33003 |
| [13] | Calinescu, C.; Milas, A.; Penn, M., Vertex algebraic structure of principal subspaces of basic \(A_{2 n}^{( 2 )}\)-modules, J. Pure Appl. Algebra, 220, 1752-1784, 2016 ·Zbl 1395.17059 |
| [14] | Cao, Z.; Rosengren, H.; Wang, L., On some double Nahm sums of Zagier, J. Comb. Theory, Ser. A, 202, Article 105819 pp., 2024 ·Zbl 1531.05016 |
| [15] | Cao, Z.; Wang, L., Multi-sum Rogers-Ramanujan type identities, J. Math. Anal. Appl., 522, 2, Article 126960 pp., 2023 ·Zbl 1516.05011 |
| [16] | Cherednik, I.; Feigin, B., Rogers-Ramanujan type identities and Nil-DAHA, Adv. Math., 248, 1050-1088, 2013 ·Zbl 1298.33029 |
| [17] | Chern, S., Asymmetric Rogers-Ramanujan type identities. I, The Andrews-Uncu conjecture, Proc. Am. Math. Soc., 151, 3269-3279, 2023 ·Zbl 1523.11188 |
| [18] | Dousse, J.; Lovejoy, J., Generalizations of Capparelli’s identity, Bull. Lond. Math. Soc., 51, 193-206, 2019 ·Zbl 1430.11144 |
| [19] | Dyson, F. J., Three identities in combinatory analysis, J. Lond. Math. Soc., 18, 35-39, 1943 ·Zbl 0028.33709 |
| [20] | Frye, J.; Garvan, F. G., Automatic proof of theta-function identities, elliptic integrals, elliptic functions and modular forms in quantum field theory, (Texts Monogr. Symbol. Comput., 2019, Springer: Springer Cham), 195-258 |
| [21] | Garoufalidis, S.; Zagier, D., Knots, perturbative series and quantum modularity ·Zbl 07897506 |
| [22] | Gasper, G.; Rahman, M., Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, vol. 96, 2004, Cambridge University Press ·Zbl 1129.33005 |
| [23] | Gordon, B., A combinatorial generalization of the Rogers-Ramanujan identities, Am. J. Math., 83, 393-399, 1961 ·Zbl 0100.27303 |
| [24] | Hardy, G. H., Lectures by Godfrey H. Hardy on the mathematical work of Ramanujan, (1937, Edwards Brothers: Edwards Brothers Ann Arbor, Michigan), Fall Term 1936. Notes taken by Marshall Hall at the Institute for Advanced Study, Princeton, NJ |
| [25] | Kanade, S.; Russell, M. C., IdentityFinder and some new identities of Rogers-Ramanujan type, Exp. Math., 24, 4, 419-423, 2015 ·Zbl 1327.11075 |
| [26] | Kanade, S.; Russell, M. C., Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type, Electron. J. Comb., 26, 1-6, 2019 ·Zbl 1409.05018 |
| [27] | Li, H.; Milas, A., Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky’s principal subspaces, J. Algebra, 640, 21-58, 2024 ·Zbl 1539.17030 |
| [28] | Mc Laughlin, J., Some more identities of Kanade-Russell type derived using Rosengren’s method, Ann. Comb., 27, 329-352, 2023 ·Zbl 1530.33017 |
| [29] | Mc Laughlin, J.; Sills, A. V.; Zimmer, P., Rogers-Ramanujan computer searches, J. Symb. Comput., 44, 8, 1068-1078, 2009 ·Zbl 1175.33018 |
| [30] | Lee, C.-H., Algebraic structures in modular q-hypergeometric series, 2012, University of California: University of California Berkeley, PhD Thesis |
| [31] | Nahm, W., Conformal field theory and the dilogarithm, (11th International Conference on Mathematical Physics (ICMP-11) (Satelite Colloquia: New Problems in General Theory of Fields and Particles). 11th International Conference on Mathematical Physics (ICMP-11) (Satelite Colloquia: New Problems in General Theory of Fields and Particles), Paris, 1994), 662-667 ·Zbl 1052.81611 |
| [32] | Nahm, W., Conformal field theory, dilogarithms and three dimensional manifold, (Nahm, W.; Shen, J.-M., Interface Between Physics and Mathematics (Proceedings, Conference in Hangzhou, People’s Republic of China, September 1993), 1994, World Scientific: World Scientific Singapore), 154-165 |
| [33] | Nahm, W., Conformal field theory and torsion elements of the Bloch group, (Frontiers in Number Theory, Physics and Geometry, II, 2007, Springer), 67-132 ·Zbl 1193.81092 |
| [34] | Paule, P., Short and easy computer proofs of the Rogers-Ramanujan identities and identities of similar type, Electron. J. Comb., 1, 1994, 9 pp., #R10 ·Zbl 0814.05009 |
| [35] | Rogers, L. J., Second memoir on the expansion of certain infinite products, Proc. Lond. Math. Soc., 25, 318-343, 1894 |
| [36] | Rogers, L. J., On two theorems of combinatory analysis and some allied identities, Proc. Lond. Math. Soc., 16, 315-336, 1917 ·JFM 46.0109.01 |
| [37] | Rosengren, H., Proofs of some partition identities conjectured by Kanade and Russell, Ramanujan J., 61, 295-317, 2023 ·Zbl 1523.11189 |
| [38] | Schur, I., Ein Beitrag zur additiven Zahlentheorie und zur Theorie der kettenbrüche, S.-B. Preuss. Akad. Wiss. Phys. Math. Kl., 302-321, 1917 ·JFM 46.1448.02 |
| [39] | Selberg, A., Über einige arithmetische Identitäten, Avh. Norske Vid.-Akad. Oslo I, 8, 1-23, 1936 ·JFM 62.1068.04 |
| [40] | Sills, A. V., An Invitation to the Rogers-Ramanujan Identities, 2018, CRC Press ·Zbl 1429.11002 |
| [41] | Slater, L. J., Further identities of the Rogers-Ramanujan type, Proc. Lond. Math. Soc. (2), 54, 1, 147-167, 1952 ·Zbl 0046.27204 |
| [42] | Stanton, D., The Bailey-Rogers-Ramanujan group, (Berndt, B. C.; Ono, K., q-Series, with Applications to Combinatorics, Number Theory, and Physics, Contemp. Math., vol. 291, 2001, American Mathematical Society: American Mathematical Society Providence, RI), 55-70 ·Zbl 1009.33016 |
| [43] | Terhoeven, M., Rationale konforme Feldtheorien, der Dilogarithmus und Invarianten von 3-Mannigfaltigkeiten, 1995, Universitat: Universitat Bonn, PhD Thesis |
| [44] | Vlasenko, M.; Zwegers, S., Nahm’s conjecture: asymptotic computations and counterexamples, Commun. Number Theory Phys., 5, 3, 617-642, 2011 ·Zbl 1256.81102 |
| [45] | Wang, L., New proofs of some double sum Rogers-Ramanujan type identities, Ramanujan J., 62, 251-272, 2023 ·Zbl 07739356 |
| [46] | Wang, L., Identities on Zagier’s rank two examples for Nahm’s problem ·Zbl 07875607 |
| [47] | Warnaar, S. O., 50 years of Bailey’s lemma, (Betten, A.; Kohnert, A.; Laue, R.; Wassermann, A., Algebraic Combinatorics and Applications, 2001, Springer: Springer Berlin, Heidelberg) ·Zbl 0972.11003 |
| [48] | Zagier, D., The dilogarithm function, (Frontiers in Number Theory, Physics and Geometry, II, 2007, Springer), 3-65 ·Zbl 1176.11026 |
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