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The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function

Christian Berg·Antonio J. Durán
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Abstract

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 $1/F(s)=1/F(s+1)-F(s+1), s>-1$.

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Christian Berg, Antonio J. Durán. 2007-02-15. The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function. https://arxiv.org/abs/math/0702446

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