IJPAM: Volume 50, No. 1 (2009)

ON FACTORIZATION OF $q$-DIFFERENCE EQUATION
FOR CONTINUOUS $q$-ULTRASPHERICAL POLYNOMIALS

I. Area$^1$, M.K. Atakishiyeva$^2$, J. Rodal$^3$
$^{1,3}$Departamento de Matemática Aplicada II
E.T.S.E. Telecomunicación
Universidade de Vigo
Campus de Vigo, As Lagoas-Marcosende, Vigo, 36310, SPAIN
$^1$e-mail: area@uvigo.es
$^3$e-mail: jrodal@edu.xunta.es
$^2$Facultad de Ciencias
Universidad Autónoma del Estado de Morelos
Cuernavaca, Morelos, C.P. 62250, MÉXICO
e-mail: mesuma@servm.fc.uaem.mx


Abstract.We prove that a customary Sturm-Liouville form of second-order $q$-difference equation for the continuous $q$-ultraspherical polynomials $C_n(x;\beta\vert\,q)$ of Rogers can be written in a factorized form in terms of some explicitly defined $q$-difference operator ${\mathcal D}_x^{\beta,\,q}$. This reveals the fact that the continuous $q$-ultraspherical polynomials $C_n(x;\beta\vert\,q)$ are actually governed by the $q$-difference equation ${\mathcal D}_x^{\beta,\,q}\, C_n(x;\beta\vert\,q)= \left(q^{-n/2}+\beta\,q^{n/2}\right)\,C_n(x;\beta\vert\,q)$, which can be regarded as a square root of the equation, obtained from its original form.

Received: October 14, 2008

AMS Subject Classification: 33D45, 39A13

Key Words and Phrases: factorization, continuous $q$-ultraspherical polynomials, $q$-difference equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 50
Issue: 1