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 q2, is a function of the Stieltjes type. Consequently, the mathematical theory of Pade approximants assures that the full function, at any finite value of q2 away from the physical cut, can be reconstructed from its low-energy power expansion around q2=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(αs2).