The Lerch zeta function as a fractional derivative

TL;DR

This paper presents a novel formulation of the Lerch zeta function as a fractional derivative, linking it to elementary functions and exploring its properties through functional equations.

Contribution

It introduces a new way to express the Lerch zeta function using fractional derivatives, enhancing understanding of its properties and relationships.

Findings

New formulation of Lerch zeta as fractional derivative

Interaction with known properties of Lerch zeta

Derivation of a second formulation via functional equation

Abstract

We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the functional equation to obtain a second formulation in terms of fractional derivatives.