Positivity preserving transformations for -binomial coefficients |
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Authors: | Alexander Berkovich S Ole Warnaar |
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Institution: | Department of Mathematics, University of Florida, Gainesville, Florida 32611 ; Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia |
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Abstract: | Several new transformations for -binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new -binomial transformations are also applied to obtain multisum Rogers-Ramanujan identities, to find new representations for the Rogers-Szegö polynomials, and to make some progress on Bressoud's generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials. |
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Keywords: | Bailey lemma base-changing transformations basic hypergeometric series Borwein conjecture $q$-binomial coefficients Rogers--Ramanujan identities Rogers--Szeg\"{o} polynomials |
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