A method for deriving hypergeometric and related identities from theH²Hardy norm of conformal maps

Abstract

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. The most easily stated of the identities we obtain are Euler's famous Basel sum, as well as the sum 1 arXiv:1205.2458v1 [math.CV]

Planar Brownian Motion and Complex Analysis

“…This is doubtless due to their definition in terms of quadratic polynomials, and the importance of quadratic polynomials in the study of Brownian motion. A set of papers containing a variety of results on Brownian motion and conics is [6,10,33,50].…”