A systematic ladder-operator approach to orthogonal polynomials

Abstract

An elementary and systematic approach to the problem of factorizing classes of second-order linear homogeneous equations, with corresponding classes of orthogonal polynomials as their solutions, is developed. The factorized formats lead to ladder-operator representations of the solution of each class of second-order linear homogeneous equation and present a different (more straightforward and explicit) resolution of the mathematical problem leading to the factorization approach of, in particular, Jafarizadeh and Fakhri [3, 4]. The main solution process involves solving coupled Riccati equations along with a third `consistency' equation directly, to find, without further assumption, all of the components of each factorization representation. As a bonus, one of the assumptions of the original Jafarizadeh and Fakhri approach is eliminated. In some cases, a basic technique of Cotfas [1, 2] is also invoked to produce simplified derivations. The main results are presented in tabular form and a brief discussion of previous/other approaches given.