On Hurwitz zeta function and Lommel functions

We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function ζ(s,a) beginning with Hermite's formula. The aim is to reveal a nice connection between ζ(s,a) and a special case of the Lommel function Sμ,ν(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.