## Taiwanese Journal of Mathematics

### 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.

#### Article information

Source
Taiwanese J. Math., Volume 8, Number 4 (2004), 593-628.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500407706

Digital Object Identifier
doi:10.11650/twjm/1500407706

Mathematical Reviews number (MathSciNet)
MR2105554

Zentralblatt MATH identifier
1074.33007

#### Citation

Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung. MULTIPLICATIVE RENORMALIZATION AND GENERATING FUNCTIONS II. Taiwanese J. Math. 8 (2004), no. 4, 593--628. doi:10.11650/twjm/1500407706. https://projecteuclid.org/euclid.twjm/1500407706

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