A method for deriving hypergeometric and related identities from the H² Hardy 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

We will be able to obtain the following hypergeometric reduction:

A related identity is

We will obtain two families of identities depending on a parameter, representative examples of which are

and

where C(k) is the kth Catalan number. We will also sum two series whose terms are defined by certain recurrence relations, and discuss an extension of the method to maps which are not conformal.

Keywords:

• hypergeometric identities
• conformal maps
• hardy spaces
• harmonic functions

Acknowledgements

The author thanks George Markowsky and Mark Coffey for useful conversations, as well as an anonymous referee for helpful comments. He is also grateful for support from Australian Research Council Grant DP0988483.