  We explore a method which is implicit in a paper of Burkholder of identifying
the $H^2$ Hardy norm of a conformal map with the explicit solution of
Dirichlet's problem in the complex plane. Using the series form of the Hardy
norm, we obtain an identity for the sum of a series obtained from the conformal
map. We use this technique to evaluate several hypergeometric sums, as well as
several sums that can be expressed as convolutions of the terms in a
hypergeometric series.
