Gamma Function and Its Functional Equations

摘要

In this article, we look at some of the less explored aspects of the gamma function. We provide a new proof of Euler’s re?ection formula and discuss its signi?cance in the theory of special functions. We also discuss the solution of Landau to a problem posed by Legendre, concerning the determination of values of the gamma function using functional identities. In 1848, Oscar Schlomilch gave an interesting additive analogue of the duplication formula. We prove a generalized version of this formula using the theory of hypergeometric functions.