Asymptotic analysis of three-parameter Mittag-Leffler function with large parameters, and application to sub-diffusion equation involving Bessel operator

In this paper, we apply the steepest descent method to the Schläfli-type integral representation of the three-parameter Mittag-Leffler function (well-known as the Prabhakar function). We find the asymptotic expansions of this function for its large parameters with respect to the real and complex saddle points. For each parameter, we separately establish a relation between the variable and parameter of function to determine the leading asymptotic term. We also introduce differentiations of the three-parameter Mittag-Leffler functions with respect to parameters and modify the associated asymptotic expansions for their large parameters. As an application, we derive the leading asymptotic term of fundamental solution of the time-fractional sub-diffusion equation including the Bessel operator with large order.