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Characterizations of the symmetric \(T_{(\theta, q)}\)-classical orthogonal \(q\)-polynomials. (English) Zbl 1514.33008

This paper is about two characterizations for symmetric \(q\)-Dunkl-classical orthogonal polynomial sequences. First, the authors show that these polynomials can be characterized as eigenfunctions of a second-order linear \(T_{(\theta, q)}\)-differential equation and, secondly, the authors characterize the linear forms as solutions of a \(T_{(\theta, q)}\)-Pearson equation (cf. Theorem 3.1). Some illustrative examples are provided. Also, the authors show that the unique \(T_{(\theta, q)}\)-classical symmetric monic orthogonal polynomial sequences are, up a dilation, the \(q^2\)-analogue of generalized Hermite polynomials and the \(q^2\)-analogue of generalized Gegenbauer polynomials (cf. Theorem 4.2).

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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