Infinite order linear differential equation satisfied by \(p\)-adic Hurwitz-type Euler zeta functions (Q2238300)

It is known, that the classical Riemann zeta function satisfies a differential equation of the form \[ T\left[\zeta(s)-1\right]=\frac{1}{s-1} \] with a certain differential operator \(T\). Also, a similar formula exists for the Hurwitz zeta function: \[ T\left[\zeta(s, a)-\frac{1}{a^{s}}\right]=\frac{1}{(s-1) a^{s-1}}\tag{*}. \] In the present paper, the authors prove that there is a \(p\)-adic version of the operator \(T\) such that the \(p\)-adic Hurwitz zeta satisfies a differential equation similar to the one presented in (*).