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Summation Chains of Sequences Part 2: Relationships between Sequences via Summation Chains

# Summation Chains of Sequences Part 2: Relationships between Sequences via Summation Chains

### Abstract

Editor’s note: This paper represents the second installment of a masters thesis by Jonathan Johnson. This paper continues the development of the theory of summation chains of sequences. The concept of sum-related is defined: two sequences are sum-related if one sequence appears in the summation chain of the other. The main result is a theorem to determine if two given sequences are sum-related.

### Introduction

Editor’s note: Previously, the concept of a summation chain of sequences was defined. We reproduce the example and introduction here:

Given a complex-valued sequence $(a_n)^{\infty}_{n=1}$, the sequence of partial sums of $(a_n)$ is given by the sequence

$$(a_1,a_1+a_2,a_1+a_2+a_3,\ldots,\sum^n_{i=1}a_i,\ldots).$$

The sequence of differences of $(a_n)$ is given by the sequence

$$(a_1, a_2-a_1,a_3-a_2,\ldots,a_n-a_{n-1},\ldots).$$

The processes of finding the sequence of partial sums and finding the sequence of differences of a sequence are inverses of each other so every sequence is the sequence of differences of its sequence of partial sums and the sequence of partial sums of its sequence of differences. Every sequence has a unique sequence of partial sums and a unique sequence of differences so it is always possible to find the sequence of partial sums of the sequence of partial sums and repeat the process ad infinitum. Similarly, we can find the sequence of differences of the sequence of differences and repeat ad infinitum. The result is a doubly infinite sequence or “chain” of sequences where each sequence is the sequence of partial sums of the previous sequence and the sequence of differences of the following sequence.

Example
Let $a^{(0)}$ be the sequence defined by $a^{(0)}_n=(-1)^{n+1}$ for all $n\in\mathbb{N}$. For all integers $m>0$, let $a^{(m)}$ be the sequence of partial sums of $a^{(m-1)}$, and for all integers $m<0$, let $a^{(m)}$ be the sequence of differences of $a^{(m+1)}$.

$$\begin{array}{r|ccccccccc}\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\a^{(2)} & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 & \cdots\\a^{(1)} & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & \cdots\\a^{(0)} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & \cdots\\a^{(-1)} & 1 & -2 & 2 & -2 & 2 & -2 & 2 & -2 & \cdots\\a^{(-2)} & 1 & -3 & 4 & -4 & 4 & -4 & 4 & -4 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\\end{array}$$

Every sequence can be used to create a unique chain of sequences.

Editor’s note: This installment examines the relationship between sequences and their summation chains, and provides a way to determine whether two sequences are sum-related.