Étude théorique et numérique de la fonction de Karlin-McGregor
type: Item
instance of: scholarly article
title: Étude théorique et numérique de la fonction de Karlin-McGregor
zbMATH Open document ID: 0545.65084
DOI: 10.1007/BF02786869
author: Serge Dubuc
published in: Journal d'Analyse Mathématique
publication date: 1983-01-01T00:00:00Z
review text: For the sake of this summary, we consider an entire function f(z) which has the following properties: If x is a real value of (0,1), then \(0<f(x)<x,\) moreover the value f'(1) is b, \(b>1\) and the value \(f'(0)\) is a, \(0<a<1\). \textit{S. Karlin} and \textit{J. McGregor} [Trans. Am. Math. Soc. 132, 115-136 and 137-145 (1968; Zbl 0159.461 and Zbl 0157.125)] introduced a function K(y) through two other functions, a Schröder's function S(x) and a Poincaré's function P(y). S(x) is analytic at 0, \(S'(0)=1\) and satisfies the functional equation \(S(f(x))=aS(x)\) for x in (0,1). P(y) is analytic at 0, \(P(0)=1, P'(0)=-1\) and satisfies the functional equation \(P(by)=f(P(y))\) for y in \((0,\infty)\). If \(c=\)- log(a)/log(b), then K(y) is defined as \(S(P(y))y^ c.\) According to the deep study of Karlin-Mc Gregor, K(y) is never a constant function. Our concern is the numerical study of the function K, which is well defined on a sector of the complex plane. In many cases, on the positive real axis, the function is almost constant. For example, if \(f(x)=(x+x^ 2)/2,\) then for any positive y, the thirteen first decimal digits of K(y) are always the same. In this paper, we undertake the numerical computation of the function S, P and K. The Fourier expansion of \(K(b^ t)\) is given and an explanation of very small variations of K on the positive axis is presented.
Mathematics Subject Classification ID: 65Q05
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 33E99
zbMATH DE Number: 3868562
zbMATH Keywords: continuous iteration
zbMATH Keywords: entire function
zbMATH Keywords: Schröder's function
zbMATH Keywords: Poincaré's function
zbMATH Keywords: functional equation
zbMATH Keywords: Fourier expansion
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3868562