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

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 ) = 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 f ( z ) = lim n → ∞ ψ ∘ n ( λ n ( λ n + 1 / λ n ) z ) , where the numbers λ n are defined by λ 0 = 0 and the recursion λ n + 1 = ( 1 / 2 ) ( λ n + λ n 2 + 4 ) . The numbers m n = 1 / λ n + 1 form a Hausdorff moment sequence of a probability measure μ such that ∫ t z - 1 d μ ( t ) = 1 / f ( z ) .