Perturbations of Laguerre–Hahn functional: modification by the derivative of a Dirac delta

Abstract

In this paper, we consider a perturbation of a Laguerre–Hahn functional by adding a derivative of a Dirac delta. This transformation leaves invariant the family of Laguerre–Hahn linear functionals. We shall also analyse the class of the perturbed linear functional. Finally, we illustrate these perturbations by adding the derivative of a Dirac delta to the first kind associated functional of the classical Laguerre functional. The expression of the new orthogonal polynomials is obtained.

Keywords:

• orthogonal polynomials
• Laguerre polynomials
• Laguerre–Hahn polynomials
• derivative of a Dirac delta

AMS Subject Classification :

• 33C47

Acknowledgements

The work of the first and second author have been supported by Comunidad de Madrid-Universidad Carlos III de Madrid, grant CCG07-UC3M/ESP-3339. The work of the second author has been supported by Dirección General de Investigación, Ministerio de Educación y Ciencia of Spain, grant MTM2006-13000-C03-02