Fourier transform and distributional representation of the gamma function leading to some new identities

We present a Fourier transform representation of the gamma functions, which leads naturally to a distributional representation for them. Both of these representations lead to new identities for the integrals of gamma functions multiplied by other functions, which are also presented here. 1. Introduction. Considering the time since Euler extended the domain of the fac-torial function from the natural numbers, N, to C by defining the gamma function [1, page 1, (1.1)] and its extensive application to a wide variety of problems [1, pages 357– 433], one would not expect to find any significant new statements about the gamma function. However, to the best of our knowledge, there is no distributional representation of this function. We present a Fourier transform representation that leads naturally to a distributional representation. Both these representations lead to new identities for the integrals of products of gamma functions with other functions, which are not included in the standard lists of properties of the gamma function [2, 3, 4].