Vol. 198, No. 2, 2001

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The spectrum of an integral operator in weighted L2 spaces

Mourad E.H. Ismail and Plamen C. Simeonov

Vol. 198 (2001), No. 2, 443–476
Abstract

We find the spectrum of the inverse operator of the q-difference operator Dq,xf(x) = (f(x) f(qx))(x(1 q)) on a family of weighted L2 spaces. We show that the spectrum is discrete and the eigenvalues are the reciprocals of the zeros of an entire function. We also derive an expansion of the eigenfunctions of the q-difference operator and its inverse in terms of big q-Jacobi polynomials. This provides a q-analogue of the expansion of a plane wave in Jacobi polynomials. The coefficients are related to little q-Jacobi polynomials, which are described and proved to be orthogonal on the spectrum of the inverse operator. Explicit representations for the little q-Jacobi polynomials are given.

Milestones
Received: 14 May 1999
Published: 1 April 2001
Authors
Mourad E.H. Ismail
Department of Mathematics
University of South Florida
Tampa, FL 33620-5700
Plamen C. Simeonov
Department of Mathematics & Computer Science
University of Houston-Downtown
Houston, TX 77002-1094