The affine group and generalized Gegenbauer polynomials

Operators of the form f(xD) — g(L), where L is a shift (lowering) operator, arise naturally in the study of stochastic processes, such as Brownian motion, on the affine group. We find the polynomial eigenfunctions and the action of the affine group as well as the matrix elements of an exponential function corresponding to L. Then it is determined when the eigenfunctions considered form a family of orthogonal polynomials.