Positivity preserving transformations for q-binomial coefficients

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--Szego polynomials, and to make some progress on Bressoud's generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on a new triple sum representations of the Borwein polynomials.