Infinite-order $p$-adic differential equations for Hurwitz-type Euler zeta functions: Higher-dimensional extensions and uniqueness of bounded solutions

Abstract

We generalize the one-dimensional infinite-order linear differential equation satisfied by the $p$-adic Hurwitz-type Euler zeta function (Abh. Math. Semin. Univ. Hambg. 91: 117--135, 2021) to $n$ variables. By encoding the differential operator as a convolution with a distribution kernel, we introduce partial zeta functions and partial operators indexed by subsets of $\{1,\dots,n\}$. A tensor product expansion of the distribution kernel is established, whose M\"obius inversion via the binomial theorem yields the higher-dimensional analogue of the original equation in the form of an alternating-sum identity, reducing to the one-dimensional case when $n=1$. Convergence of the resulting series is confirmed by non-Archimedean estimates. As a second main contribution, we prove that, under the condition $|a|_p > 2p^{1/(p-1)}$, within the Banach space of bounded analytic functions on $\mathbb Z_p$, the shifted $p$-adic Hurwitz-type Euler zeta function is the unique solution to the one-dimensional equation. This uniqueness result establishes that the infinite-order $p$-adic differential equation admits at most one bounded analytic solution, and together with the existence theorem, it characterizes the shifted $p$-adic Hurwitz-type Euler zeta function uniquely. This sharply distinguishes it from the complex setting, where the operator series diverges on the Hurwitz zeta function, and the formal kernel fails to converge in the usual spaces of analytic test functions.