Abstract

We study the spectral problem associated to a Ruijsenaars-type (q-)difference version of the one-dimensional Schrodinger operator with Poschl-Teller potential. The eigenfunctions are constructed explicitly with the aid of the inverse scattering theory of reflectionless Jacobi operators. As a result, we arrive at combinatorial formulas for basic hypergeometric deformations of zonal spherical functions on odd-dimensional hyperboloids and spheres.