The Combinatorics of q-Hermite polynomials and the Askey—Wilson Integral

Abstract

The q-Hermite polynomials are defined as a q -analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of Askey and Wilson's 4 φ 3 polynomials. A special case of this result gives the linearization formula for q -Hermite polynomials. The moments and associated continued fraction are explicitly given. Another set of polynomials, closely related to the q -Hermite, is defined. These polynomials have a combinatorial interpretation in terms of finite vector spaces which give another proof of the linearization formula and the q -analogue of Mehler's formula.