Lecture hall graphs and the Askey scheme

We establish, for every family of orthogonal polynomials in the $q$-Askey scheme and the Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall graph. This generalizes the previous results of Corteel and Kim for the little $q$-Jacobi polynomials. We build these combinatorial models by bootstrapping, beginning with polynomials at the bottom and working towards Askey–Wilson polynomials which sit at the top of the $q$-Askey scheme. As an application of the theory, we provide the first combinatorial proof of the symmetries in the parameters of the Askey–Wilson polynomials.