Polyexponentials

TL;DR

This paper introduces the polyexponential function, explores its properties, and demonstrates its applications in evaluating Mellin integrals and its connections to various special functions.

Contribution

It defines and analyzes the polyexponential, a new special function extending exponential and exponential integral functions, and links it to well-known mathematical functions.

Findings

Polyexponential extends exponential functions.

It can evaluate certain Mellin integrals.

Connections to zeta, eta, and Lerch functions are established.

Abstract

We discuss a special function (polyexponential) that extends the natural exponential function and also the exponential integral. The basic properties of the polyexponential are listed and some applications are given. In particular, it is shown that certain Mellin integrals can be evaluated in terms of polyexponentials. The polyexponential is related to the exponential polynomials, the Riemann zeta function, the Dirichlet eta function and the Lerch Transcendent.