Hurwitz型Euler zeta函数的无穷阶$p$-adic微分方程：高维推广与有界解的唯一性
Infinite-order $p$-adic differential equations for Hurwitz-type Euler zeta functions: Uniqueness and higher-dimensional extensions
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中文总结 AI 辅助
将一维无穷阶p-adic微分方程推广至n维，证明平移后的p-adic Hurwitz型Euler zeta函数在特定条件下是唯一有界解析解，区别于复情形。
中文摘要 AI 辅助
我们将$p$-adic Hurwitz型Euler zeta函数所满足的一维无穷阶线性微分方程（Abh. Math. Semin. Univ. Hambg. 91: 117--135, 2021）推广到$n$个变量。通过将微分算子编码为与分布核的卷积，我们引入了由$\{1,\dots,n\}$的子集索引的部分zeta函数和部分算子。建立了分布核的张量积展开，通过二项式定理的Möbius反演，得到原方程的高维类比，形式为交错和恒等式，当$n=1$时退化为单变量情形。所得级数的收敛性由非阿基米德估计确认。作为第二个主要贡献，我们证明在条件$|a|_p > 2p^{1/(p-1)}$下，在$\mathbb Z_p$上有界解析函数的Banach空间中，平移后的$p$-adic Hurwitz型Euler zeta函数是一维方程的唯一解。该唯一性结果确立了无穷阶$p$-adic微分方程至多有一个有界解析解，结合存在性定理，它唯一刻画了平移后的$p$-adic Hurwitz型Euler zeta函数。这将其与复解析情形鲜明区分开来，在复情形中，算子级数在Hurwitz zeta函数上发散，且形式核在通常的解析检验函数空间中不收敛。
英文摘要
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 Möbius 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.
发表机构
- South China University of Technology（华南理工大学）
- Kyungnam University（庆南大学）
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