New properties of the Lerch's transcendent

TL;DR

This paper introduces a new representation of Lerch's transcendent that enables a power series expansion valid for |z|>1, relating values at z and 1/z, enhancing understanding of its properties.

Contribution

It presents a novel representation of Lerch's transcendent for positive integer s, establishing a relation between Phi(z,n,a) and Phi(1/z,n,1-a) and deriving a convergent power series expansion.

Findings

Derived a new representation for Phi(z,n,a)

Established a relation between Phi(z,n,a) and Phi(1/z,n,1-a)

Provided a convergent power series expansion for |z|>1

Abstract

A new representation of the Lerch's transcendent Phi(z,s,a), valid for positive integer s=n=1,2,... and for z and a belonging to certain regions of the complex plane, is presented. It allows to write an equation relating Phi(z,n,a) and Phi(1/z,n,1-a), which provides an expansion of Phi(z,n,a) as a power series of 1/z, convergent for |z|>1.