On the Fourier transform of the products of M-Wright functions

Abstract

In this  note, by applying the Bromwich's integral for the inverse Mellin transform we find a new integral representation for the   M-Wright function  $$ M_\alpha(x)=\sum _{k=0}^{\infty }\frac{(-x)^{k} }{k!\Gamma (-\alpha k+1-\alpha )},\quad  \alpha=\frac{1}{2n+1}, n\in \mathbb{N},$$ and state the Fourier transform of this function. Also, using the new integral representations for the products of the M-Wright functions, we get the Fourier transform of it.