Multiplicative renormalization and generating functions. I.
type: Item
instance of: scholarly article
title: Multiplicative renormalization and generating functions. I.
zbMATH Open document ID: 1043.33004
author: Nobuhiro Asai
author: Izumi Kubo
author: Hui-Hsiung Kuo
published in: Taiwanese Journal of Mathematics
publication date: 2003-01-01T00:00:00Z
review text: Let \(\mu\) be a probability measure on \(\mathbb{R}\) with finite moments of all orders. Then there exists a sequence of polynomials \(P_{n}(x)\) which are orthogonal in \(L^{2}(\mu)\). In this paper the authors introduce a method of finding a generating function \(\psi(t,\,x)=\sum_{n=0}^{\infty}P_{n}(x)t^{n}\) for the sequence \(P_{n}(x)\). This function \(\psi(t,\,x)\) has the form \(\frac{\varphi(t,\,x)}{E_{x}\varphi(t,\,x)}\), where \(\varphi(t,\,x)\) is an appropriate function and \(E_{x}\) denotes the expectation in the \(x\)-variable with respect to the distribution \(\mu\). The construction of the function \(\varphi(t,\,x)\) is based on a result whose proof is given in a subsequent paper of the authors which is going to appear in the same journal. Several examples of known distributions are considered and the proposed method yields many sequences of classical orthogonal polynomials.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 1926037
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: generating functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500407519
OpenAlex ID: W4245062686
DOI: 10.11650/TWJM/1500407519
description: scientific article; zbMATH DE number 1926037