Integral representations and asymptotic behaviour of a Mittag-Leffler type function of two variables

TL;DR

This paper derives integral representations and asymptotic behaviors of a two-variable extension of the Mittag-Leffler function, enhancing understanding of its properties for large argument values.

Contribution

It introduces a new two-variable Mittag-Leffler type function, providing its integral representations and asymptotic formulas, which were not previously established.

Findings

Derived integral representations for the two-variable function.

Established asymptotic expansion formulas for large arguments.

Provided theorems on the function's asymptotic properties.

Abstract

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases.The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas.