Lowering operators associated with D-Laguerre–Hahn polynomials

Abstract

In this paper, a new lowering operator D _u with a linear functional u as a parameter is introduced in the linear space ℙ of all polynomials in one variable with complex coefficients. It is given by

The concept of D _u -semiclassical polynomial sequence is defined. We show that such a sequence belongs to the family of D-Laguerre–Hahn polynomials. This allows us to provide some characterizations of a D _u -semiclassical polynomial sequence in terms of a distributional equation that the linear functional satisfies as well as a structure relation. An illustrative example is considered.

Keywords:

• linear functionals
• quasi-orthogonality
• orthogonal polynomials
• Laguerre–Hahn polynomials
• Appell polynomials

AMS Subject Classifications :

• 33C45
• 42C05

Acknowledgements

The first author (F.M.) was supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, under grant MTM 2009-12740-C03-01. The work of the second author (R.S.) was supported by Faculté des Sciences de Gabès and Faculté des Sciences de Sfax, Tunisia.