Plamen Simeonov, University of Houston-Downtown, U.S.A.
e-mail: simeonov@dt.uh.edu

The Spectrum of the Inverse Operator of the q-Difference Operator

(joint work with M. E. H. Ismail)
 

Abstract

We find the spectrum of the inverse operator of the q-difference operator Dqf(x)=(f(qx)-f(x))/(x(q-1)) for a family of L2 spaces. It is shown that the spectrum is discrete and that the eigenvalues are the reciprocals of the zeros of an entire function. We obtain an expansion of the eigenfunctions in terms of big q-Jacobi polynomials which is an analog of the expansion of a plane wave in Jacobi polynomials.