  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 $\mu$, which has an increasing
and convex density on $]0,1[$, and we study some analytic functions related to
it. The Mellin transform $F$ of $\mu$ 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 $]-1,\infty[$ satisfying $F(0)=1$ and the
functional equation $1/F(s)=1/F(s+1)-F(s+1), s>-1$.
