Reflections on Eulers reflection formula and an additive analogue of Legendres duplication formula

In this note, we look at some of the less explored aspects of the gamma function. We provide a new proof of Eulers 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 Schlomilch gave an interesting additive analogue of the duplication formula. We prove a generalized version of this formula using the theory of hypergeometric functions.