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

作者：Markowsky Greg^*

来源：Integral Transforms and Special Functions, 2013, 24(4): 302-313.

DOI：10.1080/10652469.2012.689301

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

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%26apos;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%26apos;s famous Basel sum, as well as the sum %26lt;br%26gt;We will be able to obtain the following hypergeometric reduction: %26lt;br%26gt;A related identity is %26lt;br%26gt;We will obtain two families of identities depending on a parameter, representative examples of which are %26lt;br%26gt;and %26lt;br%26gt;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.

• 出版日期2013-4-1