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

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)=ψ(f(s+1)), with ψ(s)=s-1/s. Denoting ψ 1(z)=ψ(z), ψ n(z)=ψ(ψ (n-1)(z)), n 2, we prove that the meromorphic extension of f to the whole complex plane is given by the formula , where the numbers λn are defined by λ0=0 and the recursion . The numbers mn=1/λn+1 form a Hausdorff moment sequence of a probability measure μ such that .