Lecture hall graphs and the Askey scheme
arXiv:2311.12761 · doi:10.1016/j.aim.2026.111056
Abstract
We establish, for every family of orthogonal polynomials in the -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 -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 -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.
44 pages, 23 figures