Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or q-quadratic lattice

Abstract

Every classical orthogonal polynomial system $p_n(x)$ satisfies a three-term recurrence relation of the type \[ p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x)~ (n=0,1,2,\ldots, p_{-1}\equiv 0), \] with $C_nA_nA_{n-1}>0$. Moreover, Favard's theorem states that the converse is true. A general method to derive the coefficients $A_n$, $B_n$, $C_n$ in terms of the polynomial coefficients of the divided-difference equations satisfied by orthogonal polynomials on a quadratic or $q$-quadratic lattice is recalled. The Maple implementations rec2ortho of Koorwinder and Swarttouw or retode of Koepf and Schmersau were developed to identify classical orthogonal polynomials given by their three-term recurrence relation as special functions. The two implementations rec2ortho and retode do not handle classical orthogonal polynomials on a quadratic or $q$-quadratic lattice. In this manuscript, the Maple implementation retode of Koepf and Schmersau is extended to cover classical orthogonal polynomials on quadratic or $q$-quadratic lattices and to answer as application an open problem submitted by Alhaidari during the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications.