On (p, q) -classical orthogonal polynomials and their characterization theorems

Abstract

In this paper, we introduce a general (p, q) -Sturm-Liouville difference equation whose solutions are (p, q) -analogues of classical orthogonal polynomials leading to Jacobi, Laguerre, and Hermite polynomials as (p, q) → (1 , 1). In this direction, some basic characterization theorems for the introduced (p, q) -Sturm-Liouville difference equation, such as Rodrigues representation for the solution of this equation, a general three-term recurrence relation, and a structure relation for the (p, q) -classical polynomial solutions are given.