MULTIPLICATIVE RENORMALIZATION AND GENERATING FUNCTIONS II

Abstract

Let $\mu$ be a probability measure on the real line with finite moments of all orders. Suppose the linear span of polynomials is dense in $L^2(\mu)$. Then there exists a sequence $\{P_n\}_{n=0}^\infty$ of orthogonal polynomials with respect to $\mu$ such that $P_n$ is a polynomial of degree $n$ with leading coefficient $1$ and the equality $(x-\alpha_n) P_n(x) = P_{n+1}(x) + \omega_n P_{n-1}(x)$ holds, where $\alpha_n$ and $\omega_n$ are Szeg\"o-Jacobi parameters. In this paper we use the concepts of pre-generating function, multiplicative renormalization, and generating function to derive $\{P_n, \alpha_n, \omega_n\}$ from a given $\mu$. Two types of pre-generating functions are studied. We apply our method to the special distributions such as Gaussian, Poisson, gamma, uniform, arcsine, semi-circle, and beta-type to derive $\{P_n, \alpha_n, \omega_n\}$. Moreover, we show that the corresponding polynomials $P_n$'s are exactly the classical polynomials such as Hermite, Charlier, Laguerre, Legendre, Chebyshev of the first kind, Chebyshev of the second kind, and Gegenbauer. We also apply our method to study the negative binomial distributions.