从分布视角看发散级数求和的若干新例子
Some new examples of summation of divergent series from the viewpoint of distributions
AI总结：
本文从分布视角对满足条件的复数列的幂次权重发散级数和给出解释，并应用该方法推导了涉及欧拉多项式、伯努利数和Apostol–Bernoulli数的三类经典发散级数求和公式。
摘要：
设$\{a_{1}, a_{2},\ldots, a_{n},\ldots\}$是至多多项式增长且满足额外假设的复数列。受Sasane近期工作启发，本文从分布视角给出和式$a_{1}+2a_{2}+3a_{3}+\cdots+na_{n}+\cdots$的解释，更一般地，对任意$k\in\mathbb{N}$，给出和式$1^{k}a_{1}+2^{k}a_{2}+3^{k}a_{3}+\cdots+n^{k}a_{n}+\cdots$的解释。作为应用，我们解释了如下求和公式：$1^{k}-2^{k}+3^{k}-\cdots=-\frac{E_{k}(0)}{2}$，$1^{k}+2^{k}+3^{k}+\cdots=-\frac{B_{k+1}}{k+1}$，$\varepsilon^{1}1^{k}+\varepsilon^{2}2^{k}+\varepsilon^{3}3^{k}+\cdots=-\frac{B_{k+1}(\varepsilon)}{k+1}$，其中$E_{k}(0)$、$B_{k}$和$B_{k}(\varepsilon)$分别是0点的欧拉多项式、伯努利数和Apostol–Bernoulli数。
英文摘要：
Let $\{a_{1}, a_{2},\ldots, a_{n},\ldots\}$ be a sequence of complex numbers which has at most polynomial growth and satisfies an extra assumption. In this paper, inspired by a recent work of Sasane, we give an explanation of the sum $$a_{1}+2a_{2}+3a_{3}+\cdots+na_{n}+\cdots,$$ and more generally, for any $k\in\mathbb{N},$ the sum $$1^{k}a_{1}+2^{k}a_{2}+3^{k}a_{3}+\cdots+n^{k}a_{n}+\cdots,$$ from the viewpoint of distributions. As applications, we explain the following summation formulas \begin{equation*} \begin{aligned} 1^{k}-2^{k}+3^{k}-\cdots&=-\frac{E_{k}(0)}{2}, \\ 1^{k}+2^{k}+3^{k}+\cdots&=-\frac{B_{k+1}}{k+1}, \\ ε^{1}1^{k}+ε^{2}2^{k}+ε^{3}3^{k}+\cdots&=-\frac{B_{k+1}(ε)}{k+1}, \end{aligned} \end{equation*} where $E_{k}(0)$, $B_{k}$ and $B_{k}(ε)$ are the Euler polynomials at 0, the Bernoulli numbers and the Apostol--Bernoulli numbers, respectively.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
