Ultraspherical Type Generating Functions for Orthogonal Polynomials

We characterize, under some technical assumptions and up to a conjecture, probability distributions of finite all order moments with ultraspherical type generating functions for orthogonal polynomials. Our method is based on differential equations and the obtained measures are particular Beta distributions. We actually recover the free Meixner family of probability distributions so that our method gives a new approach to the characterization of free Meixner distributions. 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 positivity 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