Pade Theory applied to the vacuum polarization of a heavy quark

The vacuum polarization of a quark, when considered in terms of the external momentum q^2, is a function of the Stieltjes type. Consequently, the mathematical theory of Pade Approximants assures that the full function, at any finite value of q^2 away from the physical cut, can be reconstructed from its low-energy power expansion around q^2=0. We illustrate this point by applying this theory to the vacuum polarization of a heavy quark and obtain the value of the constant K^(2) governing the threshold expansion at order O(alpha_s^2).