The symmetric \(D_{\omega}\)-semi-classical orthogonal polynomials of class one. (English) Zbl 1181.42028
The present paper is devoted to the study of the semi-classical orthogonal polynomials of class one associated with the \(D_w\) Laguerre-Hahn form of class one, being \(D_w\) the difference operator \((D_{\omega}p)(x)= (p(x+\omega) - p(x))/\omega\). When \(w\to0\) it becomes into the derivative operator [and the corresponding problem was solved by J. Alaya and P. Maroni, Integral Transforms Spec. Funct. 4, No. 4, 301–320 (1996; Zbl 0865.42021)]. For this case (\(D_w\) case), the authors deduce the set of Laguerre-Freud equations corresponding to the \(D_w\) semi-classical orthogonal polynomials of class one and completely solve it in the symmetric case obtaining two canonical forms. This two forms are studied in details and the corresponding integral representations as well as the the parameters of the corresponding three-term recurrence of the associated family of orthogonal polynomials relation are explicitly given. By taking the limit \(\omega\to0\) they recovered their previous results (published in the aforementioned paper).
Reviewer: Renato Alvarez-Nodarse (Sevilla)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Citations:
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