MULTIPLICATIVE RENORMALIZATION AND GENERATING FUNCTIONS I.

Abstract

Let $\mu$ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system $\{1, x, x^2, \ldots, x^n, \ldots \}$ to get orthogonal polynomials $P_n(x), \,n\geq 0,$ which have leading coefficient 1 and satisfy $(x-\alpha_n)P_n(x) = P_{n+1}(x) + \omega_n P_{n-1}(x)$. In general it is almost impossible to use this process to compute the explicit form of these polynomials. In this paper we use the multiplicative renormalization to develop a new method for deriving generating functions for a large class of probability measures. From a generating function for $\mu$ we can compute the orthogonal polynomials $P_n(x),\,n\geq 0$. Our method can be applied to derive many classical polynomials such as Hermite, Charlier, Laguerre, Legendre, Chebyshev (first and second kinds), and Gegenbauer polynomials. It can also be applied to measures such as geometric distribution to produce new orthogonal polynomials.