Palindromic Riordan arrays, classical orthogonal polynomials and Catalan triangles

Abstract

We characterize the palindromic generalized Riordan arrays and their Sheffer sequences showing that, apart from a trivial case, these arrays all arise from a given array P(q,t,u) by suitably choosing the parameters q,t,u. Remarkably, well-known polynomial and numerical sequences arise as special cases of such Sheffer sequences, including classical orthogonal polynomials. After a suitable normalization, both the palindromic Sheffer sequences and their gamma polynomials have non-negative integer coefficients. We prove that these coefficients count a family of directed graphs, then through this combinatorial setting we obtain a bijective proof of the palindromic property of P(q,t,u), and we recover the combinatorial expansion of orthogonal polynomials due to F. Bergeron. Finally, we state explicit connections between the inverse Q(q,t,u) of P(q,t,u) and an array C(q,t) which generalizes Aigner's array of ballot numbers, the Pascal triangle and the Catalan triangle of Shapiro.