Iteration of the rational function z-1/z and a Hausdorff moment sequence

TL;DR

This paper characterizes a specific positive function related to Hausdorff moment sequences, providing a formula for its meromorphic extension and linking it to a probability measure via a functional equation.

Contribution

It introduces a novel explicit formula for the meromorphic extension of a function defined by a functional equation involving a rational transformation.

Findings

The function's meromorphic extension is given by a limit involving iterates of psi.

The sequence lambda_n is explicitly defined and used in the extension formula.

The sequence m_n forms a Hausdorff moment sequence of a probability measure.

Abstract

In a previous paper we considered a positive function f, uniquely determined for s>0 by the requirements f(1)=1, log(1/f) is convex and the functional equation f(s)=psi(f(s+1)) with psi(s)=s-1/s. We prove that the meromorphic extension of f to the whole complex plane is given by the formula f(z)=lim_{n\to\infty}psi^{\circ n}(lambda_n(lambda_{n+1}/lambda_n)^z), where the numbers lambda_n are defined by lambda_0=0 and the recursion lambda_{n+1}=(1/2)(lambda_n+sqrt{lambda_n^2+4}). The numbers m_n=1/lambda_{n+1} form a Hausdorff moment sequence of a probability measure \mu such that \int t^{z-1}d\mu(t)=1/f(z)