Asymptotic analysis of a family of polynomials associated with the inverse error function
type: Item
instance of: scholarly article
title: Asymptotic analysis of a family of polynomials associated with the inverse error function
zbMATH Open document ID: 1259.34047
DOI: 10.1216/RMJ-2012-42-3-847
author: Charles Knessl
author: Diego Dominici
published in: Rocky Mountain Journal of Mathematics
publication date: 2012-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0811.2243
full work available at URL: https://projecteuclid.org/euclid.rmjm/1344430862
Mathematics Subject Classification ID: 34E20
Mathematics Subject Classification ID: 33E30
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 6066664
zbMATH Keywords: inverse error function
zbMATH Keywords: differential-difference equations
zbMATH Keywords: singularity analysis
zbMATH Keywords: discrete WKB method
reviewed by: Alberto Lastra
MaRDI profile type: Publication
arXiv ID: 0811.2243
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review text: The authors present a deep study on the behavior of a family of polynomials \((P_{n})_{n\geq0}\) arising when computing higher derivatives of the inverse error function. Indeed, it is equivalent to estimate higher derivatives of the error function at 0 and the corresponding polynomial \(P_{n}(x)\) at \(x=0\) asymptotically in \(n\). The mentioned polynomials satisfy the following recurrence: NEWLINE\[NEWLINE P_0(x)\equiv1,\qquad P_{n+1}(x)=P'_{n}(x)+x(n+1)P_{n}(x),\quad n\geq1. NEWLINE\]NEWLINE The main aim of the work is to obtain different asymptotic expansions for \(n\to\infty\) and (i) \(0<x<\infty\); (ii) \(x=O(n^{-1})\); (iii) \(x=O(\sqrt{\ln(n)})\).NEWLINENEWLINEAfter a singularity analysis for \(P_n(x)\), the WKB method is applied to the previous recurrence satisfied by \((P_{n}(x))_{n\geq0}\) and the fact that \(P_{n}(x)\sim n!x^{n}\), \(x\to\infty\) in order to infer the asymptotics of \(P_n(x)\). Then, the discrete form of the ray method is used to obtain the mentioned uniform asymptotic approximations for \(P_n(x)\).NEWLINENEWLINESeveral plots comparing the estimates and the real values of \(P_n(x)\) in different cases are presented.
description: scientific article; zbMATH DE number 6066664