IJPAM: Volume 10, No. 3 (2004)

A $q$-EXTENSION OF THE GENERALIZED
HERMITE POLYNOMIALS WITH THE CONTINUOUS
ORTHOGONALITY PROPERTY ON $\RR$

R. Álvarez-Nodarse$^1$, M.K. Atakishiyeva$^2$, N.M. Atakishiyev$^3$
$^1$Departamento de Análisis Matemático
Universidad de Sevilla
Apartado Postal 1160, E-41080 Sevilla, SPAIN
and
$^1$Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada, E-18071 Granada, SPAIN
e-mail: ran@us.es
$^2$Facultad de Ciencias
UAEM - Universidad Autónoma del Estado de Moredos
Apartado Postal 396-3, C.P. 62250, Cuernavaca, Morelos, MEXICO
e-mail: mesuma@servm.fc.uaem.mx
$^3$Instituto de Matemáticas
UNAM - Universidad Nacional Autónoma de Mexico
Apartado Postal 273-3, C.P. 62210, Cuernavaca
Santa Fe 45, Col. Maravillas, Morelos, MÉXICO
e-mail: natig@matcuer.unam.mx


Abstract.In this paper we study in detail a $q$-extension of the generalized Hermite polynomials of Szego. A continuous orthogonality property on $\RR$ with respect to the positive weight function is established, a $q$-difference equation and a three-term recurrence relation are derived for this family of $q$-polynomials.

Received: November 12, 2003

AMS Subject Classification: 26C05, 33D45, 39A13

Key Words and Phrases: generalized Hermite polynomials, continuous orthogonality, $q$-difference equation, three-term recurrence relation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 10
Issue: 3