  In this paper we study a class of Hausdorff--transformed power series whose
convergence is extremely slow for large values of the argument. We perform a
Watson-type resummation of these expansions, and obtain, by the use of the
Pollaczek polynomials, a new representation whose convergence is much faster.
We can thus propose a new algorithm for the numerical evaluation of these
expansions, which include series playing a relevant role in the computation of
the partition function in statistical mechanics. By the same procedure we
obtain also a solution of the classical Hausdorff moment problem.
