On Hurwitz zeta function and Lommel functions

TL;DR

This paper presents a new proof of Hurwitz's formula for the Hurwitz zeta function, revealing a novel connection to Lommel functions and rephrasing related modular transformations.

Contribution

It introduces a new proof of Hurwitz's formula and uncovers a connection between the Hurwitz zeta function and Lommel functions, enabling alternative formulations of modular transformations.

Findings

New proof of Hurwitz's formula using Hermite's formula

Established a connection between Hurwitz zeta function and Lommel functions

Rephrased modular transformations in terms of Lommel functions

Abstract

We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function $\zeta(s, a)$ beginning with Hermite's formula. The aim is to reveal a nice connection between $\zeta(s, a)$ and a special case of the Lommel function $S_{\mu, \nu}(z)$. This connection is used to rephrase a modular-type transformation involving infinite series of Hurwitz zeta function in terms of those involving Lommel functions.