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

TL;DR

This paper extends a Maple software to identify and analyze classical orthogonal polynomials on quadratic or q-quadratic lattices, addressing an open problem and enhancing computational tools in the field.

Contribution

The authors develop an extension of the retode Maple package to handle classical orthogonal polynomials on quadratic and q-quadratic lattices, filling a gap in existing software capabilities.

Findings

Extended retode to include quadratic and q-quadratic lattices

Successfully identified classical orthogonal polynomials on these lattices

Addressed an open problem from the International Symposium on Orthogonal Polynomials

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.