发表机构
Mathematical Institute, Graduate School of Science, Tohoku University（东北大学理学研究科数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种不依赖p进积分的统一代数框架，用于判定序列的p进同余性与插值函数局部解析性，并将其应用于热诺基数等传统方法无法处理的数论序列。
AI 中文摘要
本文研究具有指数生成函数F(t)的序列{cₙ}（n从0到无穷）满足同余式c_{n+(p^h-1)p^k}≡cₙ mod p^{k+r}（n足够大时）及其规范插值函数局部解析的判定问题。这类同余常见于伯努利数、欧拉数、斐波那契数等数论或组合序列中，传统上用p进积分证明。本文提出新方法，不使用p进积分，而是研究生成函数集合的代数结构，建立F(t)满足同余及插值函数光滑性的等价或充分条件，还将该框架具体应用于热诺基数、广义欧拉数等典型p进积分方法无法覆盖的对象。
英文摘要
In this paper, we treat the problem of determining when a sequence $\{c_n\}_{n=0}^{\infty}$ with the exponential generating function $F(t)$ satisfies the congruence $c_{n+(p^h-1)p^k}\equiv c_n \pmod{p^{k+r}}$ for large $n$ and when its canonical interpolating functions are locally analytic. This type of congruence is often observed for several number-theoretic or combinatorial sequences, such as Bernoulli numbers, Euler numbers, and Fibonacci numbers. Traditionally, such congruences are proved by using $p$-adic integrations. In this paper, we present another approach to this problem. Instead of using $p$-adic integrals, we investigate an algebraic structure of the set of generating functions and establish some conditions on $F(t)$ that are equivalent to or sufficient for the congruence and smoothness of the interpolating functions. We also provide concrete applications of our framework to Genocchi numbers, generalized Euler numbers, and so on, which are outside the scope of the typical $p$-adic integration method.
Comments37 pages, 1 figure
