Reflections on Euler's reflection formula and an additive analogue of Legendre's duplication formula

TL;DR

This paper explores lesser-known aspects of the gamma function, providing new proofs and extensions of classical formulas like Euler's reflection formula and an additive analogue of Legendre's duplication formula, using hypergeometric functions.

Contribution

It offers a new proof of Euler's reflection formula, extends Landau's gamma function value determination, and generalizes Schlömilch's additive duplication formula with hypergeometric functions.

Findings

New proof of Euler's reflection formula

Extension of Landau's gamma value determination

Generalization of Schlömilch's additive duplication formula

Abstract

In this note, we look at some of the less explored aspects of the gamma function. We provide a new proof of Euler's reflection formula and discuss its significance in the theory of special functions. We also discuss a result of Landau concerning the determination of values of the gamma function using functional identities. We show that his result is sharp and extend it to complex arguments. In 1848, Oskar Schl\"omilch gave an interesting additive analogue of the duplication formula. We prove a generalized version of this formula using the theory of hypergeometric functions.