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With the help of the Mellin-Barnes transform, we show how to simultaneouslyresum the expansion of a heavy-quark correlator around q^2=0 (low-energy), q^2=4 m^2 (threshold, where m is the quark mass) and q^2=-\infty (high-energy) in asystematic way. We exemplify the method for the perturbative vector correlatorat O(alpha_s^2) and O(alpha_s^3). We show that the coefficients, Omega(n), ofthe Taylor expansion of the vacuum polarization function in terms of theconformal variable \omega admit, for large n, an expansion in powers of 1/n (upto logarithms of n) that we can calculate exactly. This large-n expansion has asign-alternating component given by the logarithms of the OPE, and a fixed-signcomponent given by the logarithms of the threshold expansion in the externalmomentum q^2.

Resummation of Threshold, Low- and High-Energy Expansions for Heavy-Quark Correlators