Summation Chains of Sequences Part 1: Introduction, Generation, and Key Definitions

# Summation Chains of Sequences Part 1: Introduction, Generation, and Key Definitions

## Abstract

(Editor’s note:) This paper represents the first installment of a masters thesis by Jonathan Johnson. This work introduces the notion of summation chains of sequences. It examines the sequence of sequences generated by partial sums and differences of terms in each level of the chain, looks at chains generated by functions, then introduces a formal definition and key formulae in the analysis of such chains.

## Introduction

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. This paper studies properties of these chains and explores the relationship between sequences and the chains they create. In particular, the following questions are investigated:

• How can the sequences in a summation chain be computed quickly? Clearly, every sequence in a chain of sequences can be computed by repeatedly finding sequences of partial sums of sequences of partial sums or finding sequences of differences of sequences of differences. It is useful, however, to be able compute any sequence in a chain given the starting sequence, $a^{(0)}$, without having to compute all the sequences in between. Methods for computing chains are discussed on Page 3.
• When do two given sequences appear in the same summation chain? When two sequences appear in the same chain, one sequence can be obtained by repeatedly finding the sequences of partial sums of sequences of partial sums of the other sequence. This process could take a long time, and it is not able to determine if two sequences do not appear in the same chain. (Editor’s note: The next installment presents a process to determine with certainty whether or not two sequences are in the same chain.)
• How much information is needed to define a summation chain? Once a chain has been computed, it appears as an array of entries. Starting with a blank array, if some numbers are added to a blank array, can they be used to define the remaining entries uniquely? (Editor’s note: Later installments explore how much information in an array of numbers is needed to determine a chain.)
• How are the convergent behaviors of sequences in a summation chain related? Can every sequence in a chain diverge? Can every sequence in a chain converge? (Editor’s note: The final installment investigates the nature of the limits of sequences in a chain.)