×

New infinite hierarchies of polynomial identities related to the Capparelli partition theorems. (English) Zbl 1475.05012

Summary: We prove a new polynomial refinement of the Capparelli’s identities. Using a special case of Bailey’s lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli’s identities. We also discuss the \(q \mapsto 1 / q\) duality transformation of the base identities and some related partition theoretic relations.

MSC:

05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions
05A19 Combinatorial identities, bijective combinatorics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ablinger, J.; Uncu, A. K., qFunctions - a Mathematica package for q-series and partition theory applications, J. Symb. Comput., 107, 145-166 (2021) · Zbl 1465.05001
[2] Alladi, K.; Andrews, G. E.; Gordon, B., Refinements and generalizations of Capparelli’s conjecture on partitions, J. Algebra, 174, 2, 636-658 (1995) · Zbl 0830.05005
[3] Andrews, G. E., Multiple series Rogers-Ramanujan type identities, Pac. J. Math., 114, 267-283 (1984) · Zbl 0547.10012
[4] Andrews, G. E.; Baxter, R. J., Lattice gas generalization of the hard hexagon model. III. q-Trinomial coefficients, J. Stat. Phys., 47, 3-4, 297-330 (1987) · Zbl 0638.10009
[5] Andrews, G. E., Schur’s theorem. Capparelli’s conjecture and q-trinomial coefficients, Contemp. Math., 166, 141-154 (1994) · Zbl 0811.05001
[6] Andrews, G. E., The Theory of Partitions, Cambridge Mathematical Library (1998), Cambridge University Press: Cambridge University Press Cambridge, Reprint of the 1976 original, MR1634067 (99c:11126) · Zbl 0996.11002
[7] Berkovich, A.; Uncu, A. K., Polynomial identities implying Capparelli’s partition theorems, J. Number Theory, 201, 77-107 (2019) · Zbl 1418.05016
[8] Berkovich, A.; Uncu, A. K., Elementary polynomial identities involving q-trinomial coefficients, Ann. Comb., 23, 3-4, 549-560 (2019) · Zbl 1431.11030
[9] Berkovich, A.; Uncu, A. K., Refined q-trinomial coefficients and two infinite hierarchies of q-series identities, (Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra. Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria) (2020), Springer: Springer Cham) · Zbl 07293158
[10] Capparelli, S., A combinatorial proof of a partition identity related to the level 3 representation of twisted affine Lie algebra, Commun. Algebra, 23, 8, 2959-2969 (1995) · Zbl 0830.17012
[11] Capparelli, S., Vertex operator relations for affine algebras and combinatorial identities (1988), Rutgers University, Ph.D. Thesis
[12] Dousse, J., On partition identities of Capparelli and Primc, Adv. Math., 370, Article 107245 pp. (2020) · Zbl 1442.05017
[13] Dousse, J.; Lovejoy, J., Generalizations of Capparelli’s identity, Bull. Lond. Math. Soc., 51, 2, 193-206 (2019) · Zbl 1430.11144
[14] Gasper, G.; Rahman, M., Basic Hypergeometric Series (2004), Cambridge University Press · Zbl 1129.33005
[15] Kanade, S.; Russell, M., Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type, Electron. J. Comb., 26, 1, Article 1.6 pp. (2019) · Zbl 1409.05018
[16] Kauers, M.; Koutschan, C., A Mathematica package for q-holonomic sequences and power series, Ramanujan J., 19, 2, 137-150 (2009), Springer · Zbl 1180.33030
[17] Koutschan, C., Advanced Applications of the Holonomic Systems Approach (September 2009), Johannes Kepler University: Johannes Kepler University Linz, RISC, Ph.D. Thesis
[18] Kurşungöz, K., Andrews-Gordon type series for Capparelli’s and Göllnitz-Gordon identities, J. Comb. Theory, Ser. A, 165, 117-138 (2019) · Zbl 1414.05042
[19] Petkovšek, M.; Wilf, H. S.; Zeilberger, D., \(A = B (1996)\), A K Peters, Ltd.: A K Peters, Ltd. Wellesley, MA, xii+212 pp · Zbl 0848.05002
[20] Paule, P.; Riese, A., A Mathematica q-analogue of Zeilberger’s algorithm based on an algebraically motivated approach to q-hypergeometric telescoping, (Special Functions, q-Series and Related Topics. Special Functions, q-Series and Related Topics, Fields Inst. Commun., vol. 14 (1997)), 179-210 · Zbl 0869.33010
[21] Paule, P., Zwei neue Transformationen als elementare Anwendungen derq-Vandermonde Formel (1982), University of Vienna, Ph.D. Thesis
[22] Riese, A., qMultiSum - a package for proving q-hypergeometric multiple summation identities, J. Symb. Comput., 35, 349-376 (2003) · Zbl 1020.33007
[23] Schneider, C., Symbolic summation assists combinatorics, Sémin. Lothar. Comb., 56, Article B56b pp. (2007) · Zbl 1188.05001
[24] Sills, A. V., On series expansions of Capparelli’s infinite product, Adv. Appl. Math., 33, 2, 397-408 (2004) · Zbl 1160.11355
[25] Tamba, M.; Xie, C. F., Level three standard modules for \(A_2^{( 2 )}\) and combinatorial identities, J. Pure Appl. Algebra, 105, 1, 53-92 (1995) · Zbl 0854.17029
[26] Warnaar, S. O., The generalized Borwein conjecture. II. Refined q-trinomial coefficients, Discrete Math., 272, 2-3, 215-258 (2003) · Zbl 1030.05004
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.