\(L^{p}\)-norms and information entropies of Charlier polynomials
type: Item
instance of: scholarly article
title: \(L^{p}\)-norms and information entropies of Charlier polynomials
zbMATH Open document ID: 1013.41012
author: Lars Larsson-Cohn
published in: Journal of Approximation Theory
publication date: 2003-04-02T00:00:00Z
review text: The Charlier polynomials \(C_n(x,a)\) are defined for \(a>0\) by \[ C_n(x,a)=\sum_{k=0}^n {n \choose k}(-a)^{n-k}x(x-1)\cdots(n-k+1). \] The author finds very accurate asymptotic formulas for the norms \[ \|C_n\|_p=\{\sum_{x=0}^\infty |f(x)|^p w(x) \}^{1/p}, \] where \(w(x)=a^xe^{-a}/x!\), \(0<p<\infty\). For example, when \(n \to \infty\), \[ \|C_n\|_3={(n!)^{2/3}n^{-2/9} \over (2\pi)^{1/3}3^{1/6}} \exp \left (n^{2/3}-{2\over 3}n^{1/3}+{4 \over 9}+ O(n^{-1/3}) \right). \] An asymptotic formula is also found for the entropy \(S_n({\widehat C}) =\sum_{x=0}^\infty {\widehat C}_n(x)^2\log {\widehat C}_n(x)^2 w(x)\) where \({\widehat C}_n=C_n/\sqrt{a^n n!}\), \[ S_n({\widehat C})=(n+a)\log(n/(ae))+3a+1-(1/2)\log 2\pi a+o(1), \qquad n \to \infty. \] Asymptotic formulas are also given for the values \(C_n(n\beta)\), with fixed \(\beta\).
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 94A17
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 1891488
zbMATH Keywords: Charlier polynomials
zbMATH Keywords: \(L_p\) norms
zbMATH Keywords: entropy
reviewed by: Yuly Makovoz
MaRDI profile type: Publication
full work available at URL: jath.2002.3691
OpenAlex ID: W2011045612
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DOI: 10.1006/JATH.2002.3691
description: scientific article; zbMATH DE number 1891488