  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).
