论文标题
从分布的角度来看，一些发散序列总和的新示例
Some new examples of summation of divergent series from the viewpoint of distributions
论文作者
论文摘要
令$ \ {a_ {1}，a_ {2}，\ ldots，a_ {n}，\ ldots \} $是一系列复数序列，最多具有多项式增长并满足额外的假设。在本文中，受Sasane的最新作品的启发，我们对总和$$ a_ {1}+2a_ {2}+3a_ {3}+3a_ {3}+\ cdots+na_ {n}+cdots，$ $ $$ 1^{k} a_ {1}+2^{k} a_ {2}+3^{k} a_ {3}+\ cdots+n^{k} a_ {n}+\ cdots，$ cdots，$$，从分布的角度来看。作为应用程序，我们说明以下求和公式\ begin {qore*} \ begin {Aligned} 1^{k} -2^{k}+3^{k} {k} - \ cdots＆= - \ frac {e_ {e_ {k} 1^{k}+2^{k}+3^{k}+\ cdots＆= - \ frac {b_ {k+1}} {k+1}，\\ ε^{1} 1^{k}+ε^{2} 2^{k}+ε^{3} 3^{k}+\ cdots＆= - \ frac {b_ {k+1}}（k+1}}（ε）}（ε）} {k+1} {k+1} {k+1} $ b_ {k} $和$ b_ {k}（ε）$分别是欧拉（Euler）多项式，分别为bernoulli编号和apostol-bernoulli数字。
Let $\{a_{1}, a_{2},\ldots, a_{n},\ldots\}$ be a sequence of complex numbers which has at most polynomial growth and satisfies an extra assumption. In this paper, inspired by a recent work of Sasane, we give an explanation of the sum $$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.
