IJPAM: Volume 16, No. 2 (2004)


M.N. Atakishiyev$^1$, V.A. Groza$^2$
$^1$Instituto de Matemáticas
Universidad Nacional Autonoma de Mexico (UNAM)
Apartado Postal 273-3
Santa Fe 45, Col. Maravillas
Cuernavaca, Morelos, 62210, MÉXICO
e-mail: mamed@matcuer.unam.mx
$^2$National Aviation University
Komarov Str. 1, Kiev, 03058, UKRAINE
e-mail: groza@i.com.ua

Abstract.We discuss an approach to $q$-Racah polynomials by means of Jacobi matrices, which represent operators of irreducible representations of the quantum algebra $U_q({\rm su}_{2})$. We diagonalize a certain Jacobi operator $I$ and show that elements of the transition matrix from the initial (canonical) basis to the basis, consisting of eigenfunctions of $I$, are expressed in terms of $q$-Racah polynomials. By using another operator $J$, we derive the orthogonality relations for $q$-Racah polynomials.

Received: August 20, 2004

AMS Subject Classification: 17B37, 33D45, 33D80

Key Words and Phrases: Jacobi matrix, $q$-Racah polynomials, orthogonality relation, quantum algebra, representation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 16
Issue: 2