Donghyun Kim: A combinatorial formula for the coefficients of the Al-Salam-Chihara polynomials

Abstract

The Al-Salam-Chihara polynomials are an important family of orthogonal polynomials in one variable x depending on 3 parameters alpha, beta and q. They are closely connected to a model from statistical mechanics called the partially asymmetric simple exclusion process (PASEP). We give a combinatorial formula for the coefficients of the (transformed) Al-Salam-Chihara polynomials and discuss the positivity phenomenon.