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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Positivity preserving transformations for $q$-binomial coefficients


Authors: Alexander Berkovich and S. Ole Warnaar
Journal: Trans. Amer. Math. Soc. 357 (2005), 2291-2351
MSC (2000): Primary 33D15; Secondary 33C20, 05E05
Published electronically: December 10, 2004
MathSciNet review: 2140441
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Abstract: Several new transformations for $q$-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 $q$-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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Additional Information

Alexander Berkovich
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: alexb@math.ufl.edu

S. Ole Warnaar
Affiliation: Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia
Email: warnaar@ms.unimelb.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03680-3
PII: S 0002-9947(04)03680-3
Keywords: Bailey lemma, base-changing transformations, basic hypergeometric series, Borwein conjecture, $q$-binomial coefficients, Rogers--Ramanujan identities, Rogers--Szeg\"{o} polynomials
Received by editor(s): April 13, 2003
Received by editor(s) in revised form: September 16, 2003
Published electronically: December 10, 2004
Additional Notes: The first author was supported in part by NSF grant DMS-0088975
The second author was supported by the Australian Research Council
Article copyright: © Copyright 2004 American Mathematical Society