A large family of \(D_w\)-semiclassical polynomials of class one. (English) Zbl 1370.33012
The authors determine all the \(L\)-semiclassical forms \(u(\beta_{0})\), (\(\beta_{0}\neq 0\)) of class \(s=1\) and \(L=D_{w}\) (the divided difference operator) through the study of the differential functional equation fulfilled by these forms and the resolution of a nonlinear system satisfied by the coefficients of the three-term recurrence relation of their sequences of monic orthogonal polynomials. First they give some properties of the \(D_{w}\)-semiclassical form of class one and using them and the Laguerre-Freud equations for the recurrence coefficients of orthogonal polynomials with respect to the \(D_{w}\)-semiclassical form of class one they obtain the required sequences.
Reviewer: Chrysoula G. Kokologiannaki (Patras)
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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