Combinatorial proofs of some limit formulas involving orthogonal polynomials

Abstract

The object of this paper is to prove combinatorially several (13 of them) limit formulas relating different families of hypergeometric orthogonal polynomials in Askey's chart classifying them. We first find a combinatorial model for Hahn polynomials which, as pointed out by Foata at the ICM (1983), “contains” models for Jacobi, Meixner, Krawtchouk, Laguerre and Charlier polynomials. Seven limit formulas are proved by “looking at surviving structures” when taking the limit. A simple model, T-structures, is then used to prove (using a different technique) four more limit formulas involving Meixner-Pollaczek, Krawtchouk, Laguerre, Charlier and Hermite polynomials. The theory of combinatorial octopuses (of F. Bergeron) is recalled and two more limits are demonstrated using new models of Meixner-Pollaczek, Laguerre, Gegenbauer and Hermite polynomials.