Abstract
In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator \(X-c\), where \(c\) is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the \(q\)-Hermite (resp. Charlier) polynomial is the only \(H_{\alpha,q}\)-classical (resp. \(\mathcal{S}_{\lambda}\)-classical) orthogonal polynomial, where \(H_{\alpha, q}:=X+\alpha H_q\) and \(\mathcal{S}_{\lambda}:=(X+1)-\lambda\tau_{-1}.\)
Highlights
Let O be a linear operator acting on the space of polynomials which sends polynomials of degree n to polynomials of degree n + n0, where n0 is a fixed integer (n ≥ 0 if n0 ≥ 0 and n ≥ |n0| if n0 < 0)
If O = D, the standard derivative, we recover the know family of classical orthogonal polynomials (Hermite, Laguerre, Bessel and Jacobi)
It is easy to see that the orthogonality is not preserved by X, we can consider and study some perturbed operators
Summary
Let O be a linear operator acting on the space of polynomials which sends polynomials of degree n to polynomials of degree n + n0, where n0 is a fixed integer (n ≥ 0 if n0 ≥ 0 and n ≥ |n0| if n0 < 0). We call a sequence {Pn}n≥0 of orthogonal polynomials O-classical if {OPn}n≥0 is orthogonal. If O = D, the standard derivative, we recover the know family of classical orthogonal polynomials (Hermite, Laguerre, Bessel and Jacobi). This characterization is called Hahn’s characterization (see [11, 18]) of the classical orthogonal polynomials. It is easy to see that the orthogonality is not preserved by X, we can consider and study some perturbed operators. We consider the following two operators (c = 0 or c = 1): Hα,q := X + αHq Sλ := (X + 1) − λτ−1,.
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