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.