The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function - Berg, Christian; Durán, Antonio j. (MATHEMATICA SCANDINAVICA, 2008)
We study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: T((a_n))_n=1/(a_0+… +a_n). We determine the corresponding measure μ, which has an increasing and convex density on ]0,1[, and we study some analytic functions related to it. The Mellin transform F of μ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on ]-1,∞[ satisfying F(0)=1 and the functional equation 1/F(s)=1/F(s+1)-F(s+1), s>-1.