### Browsed byCategory: Analysis

Summation Chains of Sequences Part 3: Sequence Chains from Linear Functions

## Summation Chains of Sequences Part 3: Sequence Chains from Linear Functions

### Abstract

(Editor’s note:) This paper represents the third installment of a masters thesis by Jonathan Johnson. The first two can be found here and here. This paper continues the development of the theory of summation chains of sequences. Since summation chains are doubly infinite, it’s important to know how little information we actually need to define a chain. The linearity of the function rules that generates a summation chain helps to answer this question. The notion of uniquely completable is defined from the set of positions, and several important theorems are developed to determine when a set of positions is uniquely completeable.

### Introduction

(Editor’s note:) In Part 2, Johnson notes that summation chains can be generated by the function rule $T_{\Sigma}: M^{\infty} \to M^{\infty}$,
$$T_{\Sigma}(x_{1}, x_{2}, x_{3},\ldots) = (x_{1}, x_{1} + x_{2}, x_{1} + x_{2} + x_{3},\ldots).$$ where $M^{\infty}$ is a $\mathbb{Z}$-Module. This is a formal way to define the operator that generates the partial sums of a given sequence. Johnson proves in this installment that this function rule is a linear operator. The linearity of these function operators assists in determining how little information we need to know about the chain to be able to uniquely define it.

Johnson introduces the notion of the set of positions, which becomes the smallest amount of information that can define a summation chain under certain conditions.

### Chains Generated by Linear Functions

Since vector-valued chains are vector-valued functions, they inherit the scalar multiplication and addition operations defined for functions on vector spaces. The linearity of the summation chain function rule leads to many interesting and useful results. The first of these is the closure of the set of summation chains under linear operations. This result holds for any function rule that is a linear operator.

#### Lemma.

Let M be a an 1-dimensional vector space with scalar field F. The summation function rule $T_{\Sigma}$ defined in Remark 3.1 of [6] is a linear operator on $M^{\infty}$.

#### Proposition.

Let $T:V\to V$ be a function rule where V is a vector space with scalar field F.
$\mathcal{C}_T$ is closed under scalar multiplication and addition if and only if T is a linear operator.

#### Corollary 1.

Let $T:V\to V$ be a bijective linear operator on vector space V with scalar field F, then $\mathcal{C}_T$ is a subspace of the space of all functions from $\mathbb{Z}$ to V, $V^{\mathbb{Z}}$, $\Phi_{\Sigma[M]}$ is an isomorphism, and $\mathcal{C}_T\cong V$.

#### Corollary 2.

Let M be a an 1-dimensional vector space with scalar field F.  $\mathcal{C}_{\Sigma[M]}$ is a vector space, and $\Phi_{\Sigma[M]}$ is an isomorphism from $M^{\infty}$ to $\mathcal{C}_{\Sigma[M]}$.

### Sets of Positions and Completeability

Viewing the summation operator as a linear operator is useful for answering the question on how little information must be known to define a chain.

Definition. A set of positions, U, is a nonempty subset of $\mathbb{Z}\times\mathbb{N}$. Let $N\in\mathbb{N}, U_N=\{(m,n)\in U:n\leq N\}$

Definition. Let M be an 1-dimensional vector space with scalar field F. Let $T:M^{\infty}\to M^{\infty}$ be a bijective linear operator. A set of positions, U, is (uniquely) completable for if for every function $d:U\to M$ there exists a (unique) T-chain, c, such that $c(m,n)=d(m,n)$ for all $(m,n)\in U$.

To say that a set of positions U is uniquely compleatable for T is the same as saying that every T-chain can be defined by defining each of the positions in U.

#### Lemma

Let M be an 1-dimensional vector space with scalar field F. Let $T:M^{\infty}\to M^{\infty}$ be a bijective linear operator. Let U be a set of positions, U is completable for T if and only if every nonempty subset of U is completable for T.

#### Lemma

Let M be an 1-dimensional vector space with scalar field F. Let $T:M^{\infty}\to M^{\infty}$ be a bijective linear operator. Let U be a set of positions, and let V be a set of positions containing U. If U is uniquely completable for T and V is completable for T, then V is uniquely completable for T.

How much information is needed to define a summation chain? The summation function rule has the additional property that it can be represented by an infinite-dimensional lower-triangular matrix. The question of how much information is need to define a summation chain will be addressed by studying matrices of this form. First, an ordering of the elements of $\mathbb{Z}\times\mathbb{N}$ is defined in order to create a consistent indexing of sets of positions.

Definition. Let $(m_1,n_1)$ and $(m_2,n_2)$ be in $\mathbb{Z}\times\mathbb{N}$, define $(m_1,n_1)\leq(m_2,n_2)$ to be the dictionary ordering $n_1 or both $n_1=n_2$ and $m_1\leq m_2$.

Remark. This definition defines a total order on $\mathbb{Z}\times\mathbb{N}$.[7]

#### Lemma

Let U be a set of positions such that $|U_N|$ is finite for all natural numbers, N, then U is well-ordered by the ordering in the above definition.

For the remainder of this section, given a set of positions, U, the elements U are assumed to be indexed by the well-ordering induced by the above definition, and $(m_i,n_i)=u_i$ for all $u_i\in U$. Given a vector space M, a linear function rule $T:M^{\infty}\to M^{\infty}$, and a set of positions U, a linear function on $M^{\infty}$ can be defined that maps a sequence $x\in M^{\infty}$ to that values of the positions in U for $\Phi_T(x)$.

Definition. Let M be an 1-dimensional vector space with scalar field F. Let $T:M^{\infty}\to M^{\infty}$ be a bijective linear operator with a lower-triangular matrix. Let U be a set of positions such that the cardinality of $U_N$ is finite for all natural numbers, N. The completion function for T with  is defined as follows.$S^T_U:M^{\infty}\to M^{|U|}$ where for all $x\in M^{\infty}$,

$$S^T_U(x)=[(T^{m_i}(x))_{n_i}]^{|U|}_{i=1}$$

The remaining results in this section show how to use completion functions to determine the completeability of sets of positions.

#### Proposition

Let M be an 1-dimensional vector space with scalar field F. Let $T:M^{\infty}\to M^{\infty}$ be a bijective linear operator with a lower-triangular matrix. Let U be a set of positions such that the cardinality of $U_N$ is finite for all natural numbers, N.

1. U is completable for T if and only if $S^T_U$ is surjective.
2. U is uniquely completable for T if and only if $S^T_U$ is bijective.

Solving completion problems utilizes many properties of infinite matrices that can be referenced in Cooke [4]. The next proposition addresses sets of positions for which a completion function is not defined.

#### Proposition

Let M be an 1-dimensional vector space with scalar field F. Let U be a set of positions such that for a natural number, N, $N<|U_N|$, then U is not completable for any bijective linear operator $T:M^{\infty}\to M^{\infty}$ with a lower-triangular matrix.

#### Proposition

Let M be an 1-dimensional vector space with scalar field F. Let U be a set of positions such that for a natural number, N, $N = |U_N|$, then U is uniquely completable for any bijective linear operator $T:M^{\infty}\to M^{\infty}$ with a lower-triangular matrix.

### Examples of Determining Completeability

Remark. Let S, T be linear transformations from vector space V to vector space W, and let $P:W\to W$ be a bijective linear operator such that $S=PV$.
$$\dim\text{Im}(S)=\dim\text{Im}(T)$$ and S is bijective if and only if T is bijective.[4, 8]

Example 1.
$$\begin{array}{c|cccccccc}m\backslash n& 1 & 2 & 3 & 4 & 5 & 6 & 7 & \cdots\\\vdots & & & & & & &\\2& & & & & & &\\1 & & & \bigcirc & & \bigcirc & & \bigcirc\\0& & & \bigcirc & & \bigcirc & & \bigcirc\\-1& & & & & & &\\\vdots & & & & & & &\end{array}$$

Let $T:\mathbb{R}^{\infty}\to\mathbb{R}^{\infty}$ where for $(x_n)\in\mathbb{R}^{\infty}$, $T(x_n)=(y_n)$ defined as follows:
$$y_n= \left\{\begin{array}{ll}x_1& \text{ if } n=1\\x_3-x_1&\text{ if } n=3\\x_n&\text{ if } n\text{ is even}\\x_n-x_{n-3}&\text{ if } n>4 \text{ and } n \text{ is odd}\\\end{array}\right.$$ Let $U=\{(0,n),(1,n):n\in\mathbb{N}, n>2, n \text{ is odd }\}$.
$$\mathcal{M}(S^T_U)=\left( \begin{array}{cccccccc}0 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\1 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 1 & 0 & 0 & \cdots\\0 & 1 & 0 & 0 & 1 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 0 & 0 & 1 & \cdots\\0 & 0 & 0 & 1 & 0 & 0 & 1 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\end{array}\right)$$

Let $A$ be the matrix row-equivalent to $\mathcal{M}(S^T_U)$ defined as follows.
\begin{aligned}A&=\left( \begin{array}{ccccccccc}-1 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 0& -1 & 1 & 0 & 0 & \cdots \\1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 0 & -1 & 1 & \cdots \\0 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\end{array}\right)\left( \begin{array}{cccccccc}0 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\1 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 1 & 0 & 0 & \cdots\\0 & 1 & 0 & 0 & 1 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 0 & 0 & 1 & \cdots\\0 & 0 & 0 & 1 & 0 & 0 & 1 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \end{array}\right)\\&=Id\\&=\left( \begin{array}{cccccccc}0 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\1 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 1 & 0 & 0 & \cdots\\0 & 1 & 0 & 0 & 1 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 0 & 0 & 1 & \cdots\\0 & 0 & 0 & 1 & 0 & 0 & 1 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\end{array}\right)\left( \begin{array}{ccccccccc}-1 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 0 & -1 & 1 & 0 & 0 & \cdots \\1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 0 & -1 & 1 & \cdots \\0 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots &\vdots\end{array}\right)\\\end{aligned} $S^T_U$ has an inverse so $S^T_U$ is bijective.Therefore, it is uniquely completable for T.

Example 2.
$$\begin{array}{c|cccccccc}m\backslash n& 1 & 2 & 3 & 4 & 5 & 6 & 7 & \cdots\\\vdots & & & & & & &\\2& & & & & & &\\1 & & \bigcirc &\bigcirc & & & & & \\0 & & & & \bigcirc & & & &\\-1& & & & \bigcirc & & & &\\2& & & & & & &\\\vdots & & & & & & &\end{array}$$

Let T be the summation operator on $\mathbb{R}^{\infty}$ with a set of positions $U=\{(1,2),(1,3),(-1,4),(0,4)\}$.

$$\mathcal{M}(S^T_U)=\left(\begin{array}{cccccccc}1 & 1 & 0 & 0 & 0 & 0 & \cdots\\1 & 1 & 1 & 0 & 0 & 0 & \cdots\\0 & 0 & -1 & 1 & 0 & 0 & \cdots\\0 & 0 & 0 & 1 & 0 & 0 & \cdots\end{array}\right)$$

Let $A$ be the matrix row-equivalent to $\mathcal{M}(S^T_U)$ defined as follows.

\begin{aligned}A&=\left(\begin{array}{cccc}1 & 0 & 0 & 0 \\-1 & 1 & 1 & -1 \\0 & 0 & -1 & 1 \\0 & 0 & 0 & 1 \end{array}\right)\left(\begin{array}{cccccccc}1 & 1 & 0 & 0 & 0 & 0 & \cdots\\1 & 1 & 1 & 0 & 0 & 0 & \cdots\\0 & 0 & -1 & 1 & 0 & 0 & \cdots\\0 & 0 & 0 & 1 & 0 & 0 & \cdots\end{array}\right)\\&=\left(\begin{array}{cccccccc}1 & 1 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 1 & 0 & 0 & 0 & \cdots\\0 & 0 & 0 & 1 & 0 & 0 & \cdots\end{array}\right)\end{aligned} $\dim\text{Im}(S^T_U)=\text{rank}(A)=3<4$ so $S^T_U$ is not surjective. Therefore, U is not completable for T.

Example 3.
$$\begin{array}{c|cccccccc}m\backslash n& 1 & 2 & 3 & 4 & 5 & 6 & 7 & \cdots\\\vdots & & & & & & &\\4& & & & & & & &\\3& & & & & &\bigcirc & &\\2 & & & & \bigcirc & & & &\\1 & & \bigcirc & & & & & & \\0&&&&&& & &\\-1 & & \bigcirc & & & & & &\\-2 & & & & \bigcirc & & & &\\-3& & & & & &\bigcirc & &\\-4& & & & & & & &\\\vdots & & & & & & &\end{array}$$

Let T be the summation operator on $\mathbb{R}^{\infty}$ with a set of positions $U=\{(n,2n),(-n,2n):n\in\mathbb{N}\}$.

$$\mathcal{M}(S^T_U)=\left(\begin{array}{ccccccccccc}1 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & \cdots\\-1 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & \cdots\\4 & 3 & 2 & 1 & 0 & 0 & 0 &\cdots & 0 & 0 & \cdots\\0 & 2 & -2 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 & \cdots\\21 & 15 & 10 & 6 & 3 & 1 & 0 & \cdots & 0 & 0 & \cdots\\0 & 0 & -1 & 3 & -3 & 1 & 0 & \cdots & 0 & 0 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots & \\& & & & & & & \cdots & n & 1 & \cdots\\& & & & & & & \cdots & -n & 1 & \cdots\\& & & & & & & & \vdots & \vdots &\end{array}\right)\begin{array}{l}\\\\\\\\\\\\\\\leftarrow \text{ row } 2n-1\\\leftarrow\text{ row }2n\\\\\end{array}$$

Let $A$ be the matrix row-equivalent to $\mathcal{M}(S^T_U)$ defined as follows.

\begin{aligned}A&=\left(\begin{array}{cccccccc}1 & -1 & 0 & 0 & 0 & 0 & \cdots\\0 & 1 & 0 & 0 & 0 & 0 & \cdots\\0 & 0 & 1 & -1 & 0 & 0 & \cdots\\0 & 0 & 0 & 1 & 0 & 0 & \cdots\\0 & 0 & 0 & 0 & 1 & -1 & \cdots\\0 & 0 & 0 & 0 & 0 & 1 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots\end{array}\right)\mathcal{M}(S^T_U)\\&=\left( \begin{array}{ccccccccccc}2 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & \cdots\\-1 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & \cdots\\4 & 1 & 4 & 0 & 0 & 0 & 0 &\cdots & 0 & 0 & \cdots\\0 & 2 & -2 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 & \cdots\\21 & 15 & 11 & 3 & 6 & 0 & 0 & \cdots & 0 & 0 & \cdots\\0 & 0 & -1 & 3 & -3 & 1 & 0 & \cdots & 0 & 0 & \cdots\\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots & \\& & & & & & & \cdots & 2n & 0 & \cdots\\& & & & & & & \cdots & -n & 1 & \cdots\\& & & & & & & & \vdots & \vdots & \end{array}\right)\\\end{aligned}

Since $A$ is lower-triangular with no zeros on the diagonal, $A$ is the matrix of a bijective linear operator. Since $A$ is equivalent to $\mathcal{M}(S^T_U)$, $S^T_U$ is bijective.
Therefore, $U$ is uniquely completable for $T$.

### Conclusion

(Editor’s Note): This installment examined sequence chains from linear functions with the goal of looking for the minimum amount of information necessary to uniquely define a summation chain. The set of positions was defined, and the notion of uniquely completable is based on this definition. Several elegant theorems gave the conditions under which the set of positions is uniquely completable. The fourth and final installment will discuss the notion of convergence for complex-valued summation chains.

### References

1. Apostol, T., 1974, Mathematical Analysis 2nd Edition, Addison-Wesley Publishing Company
2. Brin, M. and Stuck, G., 2002, Introduction to Dynamical Systems, Cambridge University Press
3. Cameron, P. J., 1992, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press
4. Cook, R. G., 2014, Infinite Matrices and Sequence Spaces, Dover Publications, Inc.
5. Johnson, J. (2017) Summation Chains of Sequences Part 1: Introduction, Generation, and Key Definitions. AACTO, Vol. 2, Issue 2.
6. Johnson, J. (2017) Summation Chains of Sequences Part 2: Relationships Between Sequences via Summation Chains AACTO, Vol. 2, Issue 2.
7. Lee, J., 2011, Introduction to Topological Manifolds, Springer Science+Business Media LLC
8. Leon, S. J., 2002, Linear Algebra with Applications 6th Edition, Prentice Hall, Inc.
9. Scheinerman, E. R., 1996, Invitation to Dynamical Systems, Dover Publications, Inc.
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.

### Summation Chains and Sequences

#### Sum-Related Chains

Consider the following sequences with their $\Sigma$-chains.

$$s_1=(1,0,0,0,\ldots)$$ $$s_2=(1,1,1,1,\ldots)$$ $$s_3=(0,1,0,0,\ldots)$$ $\Phi_{\Sigma[\mathbb{R}]}(s_1) : \begin{array}{c|ccccc} \color{SteelBlue}{m} \backslash \color{LightSeaGreen}{n} & \color{LightSeaGreen}{1} &\color{LightSeaGreen}{2} & \color{LightSeaGreen}{3} & \color{LightSeaGreen}{4} &\color{LightSeaGreen}{\cdots}\\\color{SteelBlue}{\vdots} & \vdots & \vdots & \vdots & \vdots &\\\color{SteelBlue}{2} & 1 & 2 & 3 & 4 & \cdots\\\color{SteelBlue}{1} & 1 & 1 & 1 & 1 & \cdots\\\color{SteelBlue}{0} & 1 & 0 & 0 & 0 & \cdots\\\color{SteelBlue}{-1} & 1 &-1 & 0 & 0 & \cdots\\\color{SteelBlue}{-2} & 1 & -2 & 1 & 0 & \cdots\\\color{SteelBlue}{\vdots} & \vdots & \vdots & \vdots & \vdots &\\\end{array}\quad\Phi_{\Sigma[\mathbb{R}]}(s_2) : \begin{array}{c|ccccc} \color{SteelBlue}{m} \backslash \color{LightSeaGreen}{n} & \color{LightSeaGreen}{1} &\color{LightSeaGreen}{2} & \color{LightSeaGreen}{3} & \color{LightSeaGreen}{4} &\color{LightSeaGreen}{\cdots}\\\color{SteelBlue}{\vdots} & \vdots & \vdots & \vdots & \vdots &\\\color{SteelBlue}{2} & 1 & 3 & 6 & 10& \cdots\\\color{SteelBlue}{1} & 1 & 2 & 3 & 4& \cdots\\\color{SteelBlue}{0} & 1 & 1 & 1 & 1 & \cdots\\\color{SteelBlue}{-1} & 1 & 0 & 0 & 0 & \cdots\\\color{SteelBlue}{-2} & 1 & -1 & 0 & 0 & \cdots\\\color{SteelBlue}{\vdots} & \vdots & \vdots & \vdots & \vdots &\\\end{array}$ $$\Phi_{\Sigma[\mathbb{R}]}(s_3) : \begin{array}{c|ccccc} \color{SteelBlue}{m} \backslash \color{LightSeaGreen}{n} & \color{LightSeaGreen}{1} &\color{LightSeaGreen}{2} & \color{LightSeaGreen}{3} & \color{LightSeaGreen}{4} &\color{LightSeaGreen}{\cdots}\\\color{SteelBlue}{\vdots} & \vdots & \vdots & \vdots & \vdots &\\\color{SteelBlue}{2} & 0 & 1 & 2 & 3 & \cdots\\\color{SteelBlue}{1} & 0 & 1 & 1 & 1 & \cdots\\\color{SteelBlue}{0} & 0 & 1 & 0 & 0 & \cdots\\\color{SteelBlue}{-1} & 0 &1 &-1 & 0 & \cdots\\\color{SteelBlue}{-2} & 0 & 1 & -2 & 1 & \cdots\\\color{SteelBlue}{\vdots} & \vdots & \vdots & \vdots & \vdots &\\\end{array}$$

These chains, while distinct, contain the same pattern of numbers. The entries of $\Phi_{\Sigma[\mathbb{R}]}(s_2)$ are the entries of $\Phi_{\Sigma[\mathbb{R}]}(s_1)$ moved up.This occurs because $s_2$ is the sequence of partial sums of $s_1$. The entries of $\Phi_{\Sigma[\mathbb{R}]}(s_3)$ are the entries of $\Phi_{\Sigma[\mathbb{R}]}(s_1)$ with an extra column of zeros.

The remainder of this section focuses on these relationships.

Definition. Let M be a $\mathbb{Z}$-Module. Let $(a_n)$ and $(b_n)$ be sequences in M. $(a_n)$ is sum-related to $(b_n)$, denoted $(a_n)\sim_{\Sigma}(b_n)$, if $(a_n)$ is a sequence in $\Phi_{\Sigma} ((b_n))$.

If two sequences are sum-related, then one sequence can be obtained by finding the sequences of partial sums of the sequences of partial sums of the other sequence.

Remark. The trivial sequence $(0,0,0,\ldots)$ is sum-related only to itself.

#### Proposition.

Sum relation defines an equivalence class.

#### Proposition.

Let c and d be $\Sigma$-chains. If there exists $m\in \mathbb{Z}$ such that $c(m,n)=d(m,n)$, for all $n\in \mathbb{N}$, then c=d.

#### Proposition.

Let M be a $\mathbb{Z}$-Module. Let c be a $\Sigma$-chain in M. Let $d:\mathbb{Z}\times\mathbb{N}\to M$ be defined such that there exists $k\in \mathbb{Z}$ such that $d(m,n)=c(m+k,n)$ for every $m\in \mathbb{Z}$ and $n\in \mathbb{N}$, thend is a $\Sigma$-chain,and every sequence in c is sum-related to every sequence in d.

Definition. Let c be a $\Sigma$-chain, and let $k\in \mathbb{Z}$.The $\Sigma$-chain $c^{\,k}$ is defined for all integers m and natural numbers n, by $c^{\,k}(m,n)=c(m+k,n)$. $c^{\,k}$ is said to be the $\Sigma$-chain c shifted by k.

Remark. If $s_1$ and $s_2$ are sequences generating $\Sigma$-chains $c_1$ and $c_2$ respectively, then saying $s_1 \sim_{\Sigma} s_2$ is equivalent to saying $c_1=c_2^k$ for some integer k.

When do two sequences appear in the same summation chain? By the above definition, this is the same as asking when sequences are sum-related. Consider the sequence $(x_k)=(0,0,1,-1,1,-1,\ldots)$. The sequences sum-related to $(x_k)$ are the sequences that are in $\Phi_{\Sigma[\mathbb{R}]}((x_k))$.

$$\Phi_{\Sigma[\mathbb{R}]}((x_{k}) : \begin{array}{c|ccccccc} \color{SteelBlue}{m} \backslash \color{LightSeaGreen}{n} & \color{LightSeaGreen}{1} &\color{LightSeaGreen}{2} & \color{LightSeaGreen}{3} & \color{LightSeaGreen}{4} &\color{LightSeaGreen}{5} &\color{LightSeaGreen}{6} &\color{LightSeaGreen}{\cdots}\\\color{SteelBlue}\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &\\\color{SteelBlue}{2} & 0 & 0 & 1 & 1 & 2 & 2 & \cdots\\\color{SteelBlue}{1} & 0 & 0 & 1 & 0 & 1 & 0 & \cdots\\\color{SteelBlue}{0} & 0 & 0 & 1 & -1 & 1 & -1 & \cdots\\\color{SteelBlue}{-1} & 0 & 0 & 1 & -2 & 2 & -2 & \cdots\\\color{SteelBlue}{-2} & 0 & 0 & 1 & -3 & 4 & -4 & \cdots\\\color{SteelBlue}{\vdots} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &\\\end{array}$$

Notice that each sequence in the chain starts with two zeros followed by a one. The sequences $(0,1,10,20,30,\ldots)$ and $(0,0,4,5,6,\ldots)$ could not be sum-related to $(x_k)$ since they do not begin with this sequence. This strategy can eliminate most pairs of sequences.

#### Lemma.

Given two nontrivial sequences $(a_k)$ and $(b_k)$ such that $(a_k)\sim_{\Sigma}(b_k)$, define $N_a$ and $N_b$ to be the indices of the first nonzero terms of $(a_k)$ and $(b_k)$ respectively, then $N_a= N_b$, and $a_{N_a}=b_{N_b}$.

In general, after exploiting the above lemma, it is difficult to determine sum relation of sequences. It is not possible to exhaustively check every sequence in a chain. However, when every nonzero element in the $\mathbb{Z}$-Module has infinite order, the corollary from the previous post tells where to look for sum-related sequences.
Thus in this case, a closed form procedure for determining sum relation can be used.

#### Theorem.

Let M be a $\mathbb{Z}$-Module. such that the order of every nonzero element in M is infinite. Let $(a_n)$ and $(b_n)$ be sequences in M, and let $c_a=\Phi_{\Sigma}((a_n))$. Let $N_1$ be the index of the first nonzero term of $(a_n)$, and let $N_2$ be the index of the first nonzero term of $(b_{n})$. Thus, $(a_{n})\sim_{\Sigma}(b_{n})$ if and only if the following conditions hold.

1. $N_1=N_2$, if this is true let $N=N_1=N_2$
2. $a_N=b_N$,
3. There exists $m \in \mathbb{Z}$ such that $m\cdot a_N=b_{N+1}-a_{N+1}$,
4. $b_n=c_a(m,n), \forall n \in \mathbb{N}$.

Remark.  When the integer m exists in the theorem above, it is unique.

Example. Let $M=\mathbb{Z}$. Let $(a_n)=(1,-1,-1,1,0,0,0,0,0,\ldots)$ and $b_n=n^2$, $\forall n\in \mathbb{N}$. Let $c_a=\Phi_{\Sigma}((a_n))$.

The index of the first nonzero terms of both sequences is 1. Also, $a_1=b_1=1$ so (1) and (2) are satisfied. $b_2-a_2=5=5\cdot a_1$ so (3) holds. If there is a chain relation, $(b_n)$ will be the fifth sequence in $c_a$. (4) has already been shown for $n=1$ and $n=2$. Using formulas from the theorem, the computation for the other indices is as follows:

$$c_a(5,3)={6 \choose 4}(1)+{5 \choose 4}(-1)+{4 \choose 4}(-1)=15-5-1=9=b_3$$

For $n\geq 4$,
\begin{aligned}c_a(5,n)=&\sum^{n}_{k=1}{4+n-k \choose 4}a_k\\=&{n+3 \choose 4}(1)+{n+2 \choose 4}(-1)+{n+1 \choose 4}(-1)+{n \choose 4}(1)\\=&\frac{(n+3)(n+2)(n+1)n}{24}-\frac{(n+2)(n+1)n(n-1)}{24}\\&-\frac{(n+1)n(n-1)(n-2)}{24}+\frac{n(n-1)(n-2)(n-3)}{24}\\=&\, n^2\\=&\, b_n\\\end{aligned}

Therefore, by the theorem above, $(a_n)\sim_{\Sigma}(b_n)$.

Example
Let $M=\mathbb{R}$. Let $a_n=\frac{1}{n}$ and $b_n=n$, $\forall n\in \mathbb{N}$. Let $c_a=\Phi_{\Sigma}((a_n))$. The index of the first nonzero terms of both sequences is 1. Also, $a_1=b_1=1$ so (1) and (2) are satisfied.
However $b_2-a_2=1.5$ which cannot be obtained by multiplying an integer by $a_1$.
Therefore, $(a_n)\nsim_{\Sigma}(b_n)$.

Example

Let $M=\mathbb{Z}$. Let $a_n=n^2-3n+2$ and $b_n=-\frac{n^4+3n^3+14n^2-48n+32}{7}$, $\forall n\in \mathbb{N}$. Let $c_a=\Phi_{\Sigma}((a_n))$. The index of the first nonzero terms of both sequences is 3. Also, $a_3=b_3=2$ so (1) and (2) are satisfied.
$b_4-a_4=-6=-3\cdot a_3$ so (3) holds. If there is a chain relation, $(b_n)$ will be the third sequence down in $c_a$. However, $c_a(-3,5)=0$ while $b_5=-\frac{108}{7}$.
Therefore, $(a_n)\nsim_{\Sigma}(b_n)$.

Example
Let $M=\mathcal{Z}[X]$. Let $a_n=\sum^{n-1}_{i=0}(i+1)x^{n-i}$, $\forall n\in \mathbb{N}$, and let $b_1=x$ and $b_n=x^n-x^{n-1}$, $n\geq 2$. Let $c_a=\Phi_{\Sigma}((a_n))$. The index of the first nonzero terms of both sequences is 1.
Also, $a_1=b_1=x$ so (1) and (2) are satisfied. $b_2-a_2=(x^2-x)-(x^2+2x)=-3x=-3\cdot a_1$ so (3) holds. If there is a chain relation, $(b_n)$ will be the third sequence down in $c_a$.
(4) has already been shown for $n=1$ and $n=2$.
\begin{aligned}c_a(-3,3)&={3 \choose 1}(x)-{3 \choose 2}(x^2+2x)+{3 \choose 3}(x^3+2x^2+3x)\\&=3x+3(x^2+2x)+x^3+2x^2+3x\\&=x^3-x^2\\&=b_3\\\end{aligned}

For $n\geq 4$,

\begin{aligned}c_a(-3,n)=&\sum^{3}_{k=0}(-1)^{3-k}{3 \choose 3-k}a_{n-3+k}\\=&-{3 \choose 0}\sum^{n-4}_{i=0}(i+1)x^{n-i-3}+{3 \choose 1}\sum^{n-3}_{i=0}(i+1)x^{n-i-1}\\&-{3 \choose 2}\sum^{n-2}_{i=0}(i+1)x^{n-i-1}+{3 \choose 3}\sum^{n-1}_{i=0}(i+1)x^{n-i}\\=&-\sum^{n-1}_{i=3}(i-2)x^{n-i}+3\sum^{n-1}_{i=2}(i-1)x^{n-i}-3\sum^{n-1}_{i=1}ix^{n-i}\\ &\quad+\sum^{n-1}_{i=0}(i+1)x^{n-i}\\=&-\sum^{n-1}_{i=3}(i-2)x^{n-i}+3\sum^{n-1}_{i=3}(i-1)x^{n-i}-3\sum^{n-1}_{i=3}ix^{n-i}\\&+\sum^{n-1}_{i=3}(i+1)x^{n-i}+3x^{n-2}-3x^{n-1}-6x^{n-2}+x^n+2x^{n-1}+3x^{n-2}\\=&\sum^{n-1}_{i=3}[-(i-2)+3(i-1)-3i(i+1)]x^{n-i}+x^n-x^{n-1}\\=&\, x^n-x^{n-1}\\=&\, b_n\\\end{aligned} Therefore,$(a_n)\sim_{\Sigma}(b_n)$.

#### Chains of Sequences with Leading Zeros

Adding zeros to the beginning of a sequence adds columns of zeros to be beginning of the sequences summation chain.

#### Lemma.

Let c be a nontrivial $\Sigma$-chain in $M$. Let $N=\min\{n \in \mathbb{N}:c(0,n)\neq 0\}$. For every integer, m, and for every natural number, $n, $c(m,n)=0$.

#### Proposition.

Let M be a $\mathbb{Z}$-Module. Let c be a nontrivial $\Sigma$-chain in $M$. Let the natural number $N=\min\{n \in \mathbb{N}:c(0,n)\neq 0\}$. Let $d:\mathbb{Z}\times\mathbb{N}\to M$ be defined such that $d(m,n) = c(m,n+N+1)$ for every $m\in \mathbb{Z}$ and $n\in \mathbb{N}$. Then d is a $\Sigma$-chain.

#### Corollary

Let M be a $\mathbb{Z}$-Module, and let $(a_n)$ and $(b_n)$ be sequences in M, such that for a positive integer k, $b_{n+k}=a_n$ for all natural numbers n and $b_n=0$ when $n\leq k$. Let $c_a=\Phi_{\Sigma}((a_n))$ and $c_b=\Phi_{\Sigma}((a_b))$. For every integer m and natural number $n$, $c_b(m,n+k)=c_a(m,n)$, and $c_b(m,n)=0$ when $n< k$.

### Conclusion

Editor’s note: This paper represents the second installment of the concept of summation chains of complex valued sequences. Here, a new way to determine the relationship between two sequences was defined. The concept of sum-relation was defined, and a theorem provided a way to determine the relationship between two sequences.  Part 3 will discuss sequence chains generated by linear function rules.

### References

1. Apostol, T., 1974, Mathematical Analysis 2nd Edition, Addison-Wesley Publishing Company
2. Brin, M. and Stuck, G., 2002, Introduction to Dynamical Systems, Cambridge University Press
3. Cameron, P. J., 1992, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press
4. Cook, R. G., 2014, Infinite Matrices and Sequence Spaces, Dover Publications, Inc.
5. Johnson, J. (2017)Summation Chains of Sequences Part 1: Introduction, Generation, and Key Definitions. AACTO, Vol. 2, Issue 2.
6. Lee, J., 2011, Introduction to Topological Manifolds, Springer Science+Business Media LLC
7. Leon, S. J., 2002, Linear Algebra with Applications 6th Edition, Prentice Hall, Inc.Scheinerman, E. R., 1996, Invitation to Dynamical Systems, Dover Publications, Inc.
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.)

## Chains Generated by Functions

The summation chains we have considered are doubly infinite sequences (of sequences) in which a fixed rule determines the chain of sequences recursively in both directions. The recursive rule in the backward direction is the inverse of the rule in the forward direction. Unlike single infinite sequences, in a chain there is no starting point to begin the recursive pattern. A sequence must be chosen to “start” the chain, and could be thought of as a “seed” of the chain. Starting with a given sequence, the next sequence in the chain can be obtained by applying a rule, and the previous sequence can be obtained by applying the rule’s inverse. However, this same pattern of sequences could be generated starting with any other sequence in the chain, together with the same generating rule. In order for this process to be well-defined, the rule used must have a well-defined inverse. The following definition captures abstractly what chains are. It also generalizes chains of sequences to chains of elements of more general sets.

Definition. Let $f:X\to X$ be a bijective function on a set X. We will call f a function rule. An f-chain is any function $c:\mathbb{Z} \to X$ such that $c(m)=f(c(m-1))$ for all $m \in \mathbb{Z}$. $c(0)\in X$ is called the seed of c, and c is generated by f with seed c(0). c(m) is called the mth term of the chain.

Each term of a chain is given uniquely as $c(m)=f^{m}(c(0))$.

Definition. Let f be a function rule on X. Define $\mathcal{C}_f$ to be the set of all f-chains. The f-chain seed mapping is the function $\Phi_f :X \to \mathcal{C}_f$ where $\Phi_f (x)$ is the chain generated by the seed x.

Remark. $\Phi_f$ is bijective.

Example.
Let $f:\mathbb{Z}_5 \to \mathbb{Z}_5$ where $f(n)=2n$.

This function has five chains generated by 0, 1, 2, 3, and 4. See below. Notice that the chains generated by 1, 2, 3, and 4 contain the same pattern differing only by a shift in starting point.

##### Chain Seeded by 0
$$\begin{array}{cccccccccc}m: & \cdots & -3 & -2 & -1 & 0 & 1 & 2 & 3 & \cdots \\ c(m): & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\\end{array}$$
##### Chain Seeded by 1
$$\begin{array}{cccccccccc}m: & \cdots & -3 & -2 & -1 & 0 & 1 & 2 & 3 & \cdots \\ c(m): & \cdots & 2 & 4 & 3 & 1 & 2 & 4 & 3 & \cdots \\\end{array}$$
##### Chain Seeded by 2
$$\begin{array}{cccccccccc}m: & \cdots & -3 & -2 & -1 & 0 & 1 & 2 & 3 & \cdots \\ c(m): & \cdots & 1 & 2 & 4 & 3 & 1 & 2 & 4 & \cdots \\\end{array}$$

Definition Let f be a function rule on X. Let $x_1$ and $x_2$ be in $X$. $x_1$ is f-chain related to $x_2$, denoted $x_1 \sim_{f} x_2$, if $x_1$ is in the image of the function chain $\Phi_f (x_2)$.

#### Proposition.

An f-chain relation is an equivalence relation.

This relation is useful to find relationships between elements in the set X induced by f. In the example above, 1, 2, 3, and 4 are all f-chain related while 0 is related only to itself. This section ends with a few notable properties of chains that are useful, in particular when dealing with summation chains.

#### Proposition.

Let $c_1$ and $c_2$ be f-chains. If for any $m \in \mathbb{Z}, c_1(m)=c_2(m)$, then $c_1=c_2$.

#### Proposition.

Let X be a set with rule f. Let c be an f-chain with co-domain X, and let function $d:\mathbb{Z}\to X$ be such that for some $k\in \mathbb{Z}$, $d(m)=c(m+k)$ for all $m \in \mathbb{Z}$, then d is an f-chain. Also, for every pair $x\in \text{Im}(c)$ and $y\in \text{Im}(d)$, $x \sim_{f} y$.

A more detained analysis of sequences generated by recursive funtions in Brin and Stuck (2). The remainder of this paper will focus on summation chains of sequences.

## Summation Chains of Sequences

### Key Definition and Formulas

Let M be a set, and let X be the set of sequences of elements in M. Given a function rule f on X, and an f-chain, c, in order to keep notation clean, new notation is defined as follows. c(m,n) := the nth term of the sequence c(m).

Using this notation we give a clean definition of a summation chain of sequences.

Definition. Let M be a $\mathbb{Z}$-Module. A summation chain, denoted $\Sigma$-chain, of is a function $c:\mathbb{Z} \times \mathbb{N} \to M$ where $c(m,n)=\sum_{k=1}^{n} c(m-1,k)$, for all $m\in \mathbb{Z}$ and for all $n\in \mathbb{N}$$\Phi_{\Sigma[M]}$ is the seed mapping for summation chains in M. The notation $\Phi_{\Sigma}$ is used when the $\mathbb{Z}$-Module, M, is clear. $\mathcal{C}_{\Sigma[M]}$ denotes the set of all M-valued summation chains.

Remark. Summation chains are generated by the function rule $T_{\Sigma}:M^{\infty}\to M^{\infty}$ where $T_{\Sigma}(x_1,x_2,x_3,\ldots)=(x_1,x_1+x_2,x_1+x_2+x_3,\ldots)$.

Example. Let $t:\mathbb{Z}\times \mathbb{N} \to M$ be the constant function $t(m,n)=0$.

$$\sum_{k=1}^{n} t(m-1,k) =\sum_{k=1}^{n} 0 =0=t(m,n),$$

so t is a $\Sigma$-chain. This chain is the trivial chain and is generated with the sequence of all zeros.

How can the sequences in a summation chain be computed quickly? The next two results provide ways to compute entries of a summation chain. The next proposition provides a way to compute the entries in a chain using the other entries close to it.

#### Proposition.

Let M be a $\mathbb{Z}$-Module. Let $c:\mathbb{Z} \times \mathbb{N} \to M$ be a function. c is a $\Sigma$-chain in M if and only if for all $m,m' \in \mathbb{Z}$ and $n \in \mathbb{N}$, the following equalities hold.

1. $c(m,1)=c(m',1)$,
2. If $n \geq 2$, then
$$c(m,n)=c(m,n-1)+c(m-1,n)=c(m+1,n)-c(m+1,n-1).$$

The next theorem provides a way to calculate any sequence in a summation chain directly given any other sequence in the chain without having to compute any of the sequences in between.

#### Lemma.

Let natural numbers, n and k, where k < n.
$${n \choose k}={n-1 \choose k-1}{n-1 \choose k}$$

#### Theorem.

Let c be a $\Sigma$-chain in $\mathbb{Z}$-Module M. Let $m_1,m_2 \in \mathbb{Z}$ such that $m_1>m_2$, and let $n\in \mathbb{N}$. Define $m=m_1-m_2$.
$$c(m_1,n)=\sum_{k=1}^{n} {m+n-k-1 \choose m-1} c(m_2,k)$$

If $n\geq m+1$,

$$c(m_2,n)= \sum_{k=0}^{m} (-1)^{m-k}{m \choose m-k} c(m_1,n-m+k).$$

If $n\leq m+1$,

$$c(m_2,n)=\sum_{k=1}^{n} (-1)^{n-k}{m \choose n-k} c(m_1,k).$$

Remark.When $n=m+1$, the two equations above are the same.

Example. Let $c=\Phi_{\Sigma[\mathbb{R}]}((2^{k-1}))$. The following is a computation of the sequences c(-5,n) and c(-6,n).

c(-5,1)=c(0,1)=1.
c(-5,2)=c(0,2)-5c(0,1)=2-5(1)=-3.
c(-5,3)=c(0,3)-5c(0,2)+10c(0,1)=4-5(2)+10(1)=4.
c(-5,4)=c(0,4)-5c(0,3)+10c(0,2)-10c(0,1)=8-5(4)+10((2)-10(1)=-2.
c(-5,5)=c(0,5)-5c(0,4)+10c(0,3)-10c(0,2)+5c(0,1)=16-5(8)+10(4)-10((2)+5(1)=1.

When $n\geq 6$,

\begin{aligned}c(-5,n)&=\sum_{i=0}^4{4 \choose i}c(0,n-i)\\&=c(0,n)-5c(0,n-1)+10c(0,n-2)\\&\kern{3em}-10c(0,n-3)+5c(0,n-4)-c(0,n-5)\\&=2^{n-1}-5\cdot 2^{n-2}+10\cdot 2^{n-3}-10\cdot 2^{n-4}+5\cdot 2^{n-5}-2^{n-6}\\&=2^{n-6}(32-5\cdot 16+10\cdot 6-10\cdot 4+5\cdot 2-1)\\&=2^{n-6}(1)\\&=2^{n-6}\end{aligned}

c(-6,1)=c(-5,1)=1
c(-6,2)=c(-5,2)-c(-5,1)=-3-1=-4
c(-6,3)=c(-5,3)-c(-5,2)=4-(-3)=7
c(-6,4)=c(-5,4)-c(-5,3)=-2-4=-6
c(-6,5)=c(-5,5)-c(-5,4)=1-(-2)=3
c(-6,6)=c(-5,6)-c(-5,5)=1-1=0

When $n\geq 7, c(-6,n)=c(-5,n)-c(-5,n-1)=2^{n-6}-2^{n-7}=2^{n-7}$

Example. Let $a_1=1, a_2=-1$, and $a_k=0$ when $k\geq 3$. Let $c=\Phi_{\Sigma[\mathbb{R}]}((a_k))$. The following is a computation of the sequence c(4,n).

c(4,1)=c(0,1)=a_1=1

When $n\geq 2$,
\begin{aligned}c(4,n)&=\sum^{n}_{k=1}{3+n-k \choose 3}c(0,k)\\&={n+2 \choose 3}-{n+1 \choose 3}\\&=\frac{n(n+1)(n+2)}{6}-\frac{(n-1)n(n+1)}{6}\\&=\frac{n(n+1)[(n+2)-(n-1)]}{6}\\&=\frac{n(n+1)}{2}\\\end{aligned}

### Notable Corollaries

In general, there is no way to determine the exact distance between two sequences that occur in a summation chain.
Under certain conditions, however, the corollary below determines the location of sequences in relation to each other within a summation chain. This corollary is essential in determining whether two arbitrary sequences appear in the same summation chain.

#### Corollary

Let M be a $\mathbb{Z}$-Module such that the order of every nonzero element in M is infinite. Let c be a non-trivial $\Sigma$-chain in M. Let $N=\min\{n \in \mathbb{N}\::\:c(0,n)\neq 0\}$. Let $m^*,m_1,m_2 \in \mathbb{Z}$.
$$m^*\cdot c(m_1,N)=c(m_2,N+1)-c(m_1,N+1)$$

if and only if $m^*=m_2-m_1$.

Remark. When M is a field with unity $1_M$, the equation above can be written,
$$m^*\cdot(1_M)=\frac{c(m_2,N+1)-c(m_1,N+1)}{c(m_1,N)}$$

In the case of complex valued summation chains, each chain never repeats a sequence unless the chain is trivial. This property comes from the fact that zero is the only complex number that has finite order as stated by the next corollary.

#### Corollary

Let M be a $\mathbb{Z}$-Module such that the order of every nonzero element in M is infinite. Let c be a $\Sigma$-chain in M. Let $m_1,m_2 \in \mathbb{Z}$ such that $m_1\neq m_2$. If $c(m_1,n)=c(m_2,n), \forall n\in \mathbb{N}$, then c is the trivial chain.

It has not been determined whether or not the corollary above is true for all summation chains of sequences.

## Conclusion

Editor’s note: This paper represents the introduction of the concept of summation chains of complex valued sequences. Key concepts and definitions that set the stage for further results in the development of this theory. Part 2 will discuss sum-related chains, and handling leading zeros.