A method for deriving hypergeometric and related identities from the $H^2$ Hardy norm of conformal maps

TL;DR

This paper introduces a novel method leveraging the $H^2$ Hardy norm of conformal maps to derive identities for hypergeometric series and related sums, connecting complex analysis with series evaluation techniques.

Contribution

It presents a new approach to obtain hypergeometric identities by linking conformal map norms with Dirichlet problem solutions, expanding tools for series evaluation.

Findings

Derived identities for hypergeometric series sums.

Evaluated several hypergeometric and convolution sums.

Connected Hardy norm techniques with series summation methods.

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.