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.

在本文中，我们将最速下降法应用于三参数 Mittag-Leffler 函数（众所周知的 Prabhakar 函数）的 Schläfli 型积分表示。我们发现该函数相对于实数和复数鞍点的大参数的渐近展开式。对于每个参数，我们分别建立函数的变量和参数之间的关系，以确定首渐近项。我们还介绍了三参数 Mittag-Leffler 函数在参数方面的微分，并修改了其大参数的相关渐近展开式。作为一个应用，我们推导了包含大阶 Bessel 算子的时间分数次扩散方程基本解的首渐近项。