On a generalized three-parameter Wright function of the Le Roy type

TL;DR

This paper introduces a generalized Wright function of the Le Roy type, providing integral representations, Laplace transform, asymptotic expansions, and analytic continuation, advancing understanding of special functions related to Mittag-Leffler functions.

Contribution

It extends the Le Roy function by deriving new integral formulas, asymptotic behavior, and continuation methods, enriching the theory of related special functions.

Findings

Derived two integral representations of the generalized Wright function.

Calculated the Laplace transform of the function.

Provided asymptotic expansion and analytic continuation for specific parameter cases.

Abstract

Recently S. Gerhold and R. Garra-F. Polito independently introduced a new function related to the special functions of Mittag-Leffler family. This function is a generalization of the function studied by E. Le Roy in the period 1895-1905 in connection with the problem of analytic continuation of power series with a finite radius of convergence. In our note we obtain two integral representations of this special function, calculate its Laplace transform, determine an asymptotic expansion of this function on the negative semi-axis (in the case of an integer third parameter $\gamma$) and provide its continuation to the case of a negative first parameter $\alpha$. An asymptotic result is illustrated by numerical calculations. Discussion on possible further studies and open questions are also presented.