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Supernomial Coefficients, Polynomial Identities and q-Series

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Abstract

q-Analogues of the coefficients of xa in the expansion ofΠ Nj=1 (1 + x + ⋯ + xj)Lj are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the “q-supernomial coefficients” are derived, and a combinatorial interpretation using generalized Durfee dissection partitions is given. Polynomial identities of boson-fermion-type, based on the continued fraction expansion of p/k and involving the q-supernomial coefficients, are proven. These include polynomial analogues of the Andrews-Gordon identities. Our identities unify and extend many of the known boson-fermion identities for one-dimensional configuration sums of solvable lattice models, by introducing multiple finitization parameters.

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Authors and Affiliations

  1. Institute for Theoretical Physics, State University of New York, Stony Brook, New York, 11794-3840. E-mail

    Anne Schilling

  2. Department of Mathematics, University of Melbourne, Parkville, Victoria, 3052, Australia. E-mail

    S. Ole Warnaar

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  1. Anne Schilling
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  2. S. Ole Warnaar
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Schilling, A., Warnaar, S.O. Supernomial Coefficients, Polynomial Identities and q-Series. The Ramanujan Journal 2, 459–494 (1998). https://doi.org/10.1023/A:1009780810189

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