Étude théorique et numérique de la fonction de Karlin-McGregor
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.
scientific article; zbMATH DE number 3868562