Fractional differential relations for the Lerch zeta function

TL;DR

This paper explores fractional derivatives of the Lerch zeta function, deriving a partial differential equation involving an infinite series of such derivatives, expanding understanding of its fractional calculus properties.

Contribution

It introduces new fractional differential relations for the Lerch zeta function and establishes a PDE involving an infinite series of fractional derivatives.

Findings

Derived a PDE satisfied by the Lerch zeta function

Expressed the Lerch zeta function as a fractional derivative

Extended fractional calculus understanding of special functions

Abstract

Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential equation, involving an infinite series of fractional derivatives, which is satisfied by the Lerch zeta function.