Ultraspherical Type Generating Functions for Orthogonal Polynomials

We characterize the probability distributions of finite all order moments having generating functions for orthogonal polynomials of ultraspherical type. 1. Motivation: Meixner families There is a one to one correspondance between probability distributions on the real line and polynomials of a one variable satisfying a three-terms recurrence relation subject to some positive conditions ([9]). That is why in most of the cases, if not all, one tries to characterize probability distributions using generating functions for orthogonal polynomials. Among the famous generating functions are the ones of exponential type, that is if μ is a probability distribution with a finite exponential moment in a neighborhood of zero ∫