Johann Cigler
Published 2016-02-25Version 1
The Rogers-Szeg\"o polynomials are natural q-analogues of Newton binomials. In general there is no closed expression for them. We study some exceptional cases which can be factored into a product of a nice factor with a closed formula and an ugly factor with nice values for q=1 and q=-1. These can be interpreted as extensions of Gauss identity for alternating sums of q-binomial coefficients. In the course of this investigation we were led to many interesting conjectures. The most notable ones are explicit formulae for Hankel determinants of normalized Rogers-Szeg\"o polynomials.
Generating Functions for Some Families of the Generalized Al-Salam-Carlitz $q$-Polynomials
Multiple Hilbert transform associated with polynomials
The kissing polynomials and their Hankel determinants
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