Positivity preserving transformations for $q$-binomial coefficients
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- by Alexander Berkovich and S. Ole Warnaar PDF
- Trans. Amer. Math. Soc. 357 (2005), 2291-2351 Request permission
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.References
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Additional Information
- Alexander Berkovich
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- MR Author ID: 247760
- Email: alexb@math.ufl.edu
- S. Ole Warnaar
- Affiliation: Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia
- MR Author ID: 269674
- Email: warnaar@ms.unimelb.edu.au
- 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 - © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2291-2351
- MSC (2000): Primary 33D15; Secondary 33C20, 05E05
- DOI: https://doi.org/10.1090/S0002-9947-04-03680-3
- MathSciNet review: 2140441