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On \((p,q)\)-classical orthogonal polynomials and their characterization theorems. (English) Zbl 1444.33007

Summary: 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 \((p, q) \rightarrow(1,1)\). 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.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
34B24 Sturm-Liouville theory
39A70 Difference operators
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