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Ultraspherical type generating functions for orthogonal polynomials

Demni N.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 2

281--296

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.