Analytic properties of Laguerre-type orthogonal polynomials

Abstract

In this paper, we consider sequences of monic polynomials orthogonal with respect to an inner product

where M∈ℝ₊, and a∈ℝ₋.

We focus our attention on the representation of these polynomials in terms of the standard Laguerre polynomials as well as hypergeometric functions. The lowering and raising operators associated with these polynomials are obtained. The distribution of their zeros is analysed in terms of their dependence of M. Finally, some outer asymptotic properties of such orthogonal polynomials are discussed.

AMS Subject Classification :

• 33C47

Keywords:

• orthogonal polynomials
• Laguerre polynomials
• hypergeometric functions
• lowering and raising operators
• zeros
• outer relative asymptotics
• Mehler–Heine formula

Acknowledgements

The work of the second and third authors has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, grant MTM2009-12740-C03-01. It was finished during a stay at the School of Mathematics of Georgia Institute of Technology. We would like to thank J.S. Geronimo and P.I. Iliev for the hospitality as well as the comments and suggestions to improve the presentation of the manuscript.