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

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 ₄φ₃ 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.