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

Abstract

Abstract In this paper, we introduce a general ( p , q ) $(p, q)$ -Sturm-Liouville difference equation whose solutions are ( p , q ) $(p, q)$ -analogues of classical orthogonal polynomials leading to Jacobi, Laguerre, and Hermite polynomials as ( p , q ) → ( 1 , 1 ) $(p, q) \to(1,1)$ . In this direction, some basic characterization theorems for the introduced ( p , q ) $(p, q)$ -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 ) $(p, q)$ -classical polynomial solutions are given.

Keywords

• ( p , q ) $(p,q)$ -Sturm-Liouville problems
• ( p , q ) $(p,q)$ -classical orthogonal polynomials
• ( p , q ) $(p,q)$ -Pearson difference equation
• ( p , q ) $(p,q)$ -integrals
• ( p , q ) $(p,q)$ -difference operators