A note on an extension of Gelfond's constant

TL;DR

This paper introduces a natural extension of Gelfond's constant using hypergeometric functions, also extending its square root, and discusses interesting special cases.

Contribution

It presents a novel extension of Gelfond's constant and its square root through hypergeometric functions, expanding the mathematical understanding of these constants.

Findings

Extended Gelfond's constant via hypergeometric functions

Derived an extension for the square root of Gelfond's constant

Presented interesting special cases of the extension

Abstract

The aim of this note is to provide a natural extension of Gelfond's constant $e^\pi$ using a hypergeometric function approach. An extension is also found for the square root of this constant. A few interesting special cases are presented.