Some new examples of summation of divergent series from the viewpoint of distributions

TL;DR

This paper interprets divergent series involving polynomially growing complex sequences using distribution theory, providing new explanations for classical summation formulas involving Bernoulli and Euler numbers.

Contribution

It introduces a distribution-based framework to interpret divergent series of polynomially growing sequences, extending recent work and connecting to classical special number formulas.

Findings

Provides distributional explanations for divergent series sums.

Derives formulas involving Bernoulli and Euler numbers.

Connects divergent series summation to special number theory.

Abstract

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}, \\ \epsilon^{1}1^{k}+\epsilon^{2}2^{k}+\epsilon^{3}3^{k}+\cdots&=-\frac{B_{k+1}(\epsilon)}{k+1}, \end{aligned} \end{equation*} where $E_{k}(0)$, $B_{k}$ and $B_{k}(\epsilon)$ are the Euler polynomials at 0, the Bernoulli numbers and the Apostol--Bernoulli numbers, respectively.