Picturesque convolution-like recurrences and partial sums' generation

TL;DR

This paper introduces methods to generate new sequences related to a known sequence using convolution-like recurrences, exploring their properties, limits, and connections to famous sequences like the partial sums of the Riemann zeta function.

Contribution

It presents novel techniques for constructing sequences via convolution-like recurrences and analyzes their asymptotic behavior and geometric patterns.

Findings

Sequences can be related through convolution-like recurrences.

Limit behavior of sequences depends on initial terms and the known sequence.

Examples include partial sums of the Riemann zeta function.

Abstract

Let ${\pmb b}=\{b_0,\,b_1,\,\ldots\}$ be the known sequence of numbers such that $b_0\neq0$. In this work, we develop methods to find another sequence ${\pmb a}=\{a_0,\,a_1,\,\ldots\}$ that is related to ${\pmb b}$ as follows: $a_n=a_0\,b_{n+m}+a_1\,b_{n+m-1}+\ldots+a_{n+m}\,b_0$, $n\in\mathbb{N}\cup\{0\}$, $m\in\mathbb{N}$. We show the connection of $\lim_{n\to\infty}a_n$ with $a_0,\,a_1,\,\ldots,\,a_{m-1}$ and provide varied examples of finding the sequence ${\pmb a}$ when ${\pmb b}$ is given. We demonstrate that the sequences ${\pmb a}$ may exhibit pretty patterns in the plane or space. Also, we show that the properly chosen sequence ${\pmb b}$ may define ${\pmb a}$ as some famous sequences, such as the partial sums of the Riemann zeta function, etc.