Positivity preserving transformations for $q$-binomial coefficients

By Alexander Berkovich, S. Ole Warnaar. 2003

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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 - Szego 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/672d153bced71b4a40dfeebdc3fef13f70df4c43