Autoconvolution equations and generalized Mittag-Leffler ‎functions

This article is devoted to study of the autoconvolution equations and generalized Mittag-Leffler functions. These types of equations are given in terms of the Laplace transform convolution of a function with itself. We state new classes of the autoconvolution equations of the first kind and show that the generalized Mittag-Leffler functions are solutions of these types of equations. In view of the inverse Laplace transform, we use the Schouten-Vanderpol theorem to establish an autoconvolution equation for the generalized Mittag-Leffler functions in terms of the Laplace and Mellin transforms. Also, in special cases we reduce the solutions of the introduced autoconvolution equations with respect to the Volterra $mu$-functions. Moreover, more new autoconvolution equations are shown using the Laplace transforms of generalized Mittag-Leffler functions. Finally, as an application of the autoconvolution equations in thermodynamic systems, we apply the Laplace transform for solving the Boltzmann equation and show its solution in terms of generalized Mittag-Leffler ‎functions.‎