Orthogonal polynomials associated with natural exponential families.

Summary This work proposes the generalization to several dimensions of different characterizations of the class of real quadratic natural exponential families, using the theory of orthogonal polynomials. In a first part, written in English and submitted for publication, we develop the three following characterizations: i) of meixner (1934) concerning the orthogonal polynomials of exponential generating function, ii) of feinsilver (1986) where the polynomials are obtained by derivations of the densities of probabilities, iii) of shanbhag (1972) where appears the bhattacharrya matrices. By introducing an original construction of orthogonal polynomials, we obtain a characterization of natural quadratic and simple quadratic multidimensional exponential families. Moreover, we determine the class of orthogonal polynomials in several variables whose generating function is exponential. The second part is inspired by an article of feinsilver (1991) which shows a link between the algebra of links and the theory of probabilities. Based on this work, we then show the existence of a bijection between the class of simple quadratic natural exponential families and three types of Lie algebras. Thus, any probability of a natural exponential family allows to define operators of one of the three lie algebras in question thanks to the recurrence equations of the orthogonal polynomials considered in the first part.