Draft:The composite function series

The composite function series is a mathematical series that places a function of two variables into a series, where each previous term is substituted into the next term. The series utilizes the mathematical concept of function composition. Unlike other mathematical series, the last term in the series appears as the first term. This process continues until $f(x,n)$ is reached.

The composite function series was introduced on April 30, 2025, in an article authored by Camden W. Hulse.

Definition

The composite function series is expressed using the following definition:

Let $k$ and $j$ be on the integers, let $n$ be the maximum value of $k$, let $f(x,k)$ be a real-valued function, and let $\circ$ be an operator where the right side is plugged into the real variable, $x$, on the left side. The first term in the series is $f(x,j)$, and the last term in the series is $f(x,n)$. For each $n\in \mathbb {Z}$ and $n\geq k$, define the composite function series as the following:

$\circ _{k=j}^{n}[f(x,k)]=f(x,n)\circ f(x,n-1)\circ f(x,n-2)\circ \ldots \circ f(x,j)$

Hulse then provides several basic examples which further describe the series:

$\circ _{k=4}^{6}[x^{k}]=x^{6}\circ x^{6-1}\circ x^{6-2}=x^{6}\circ x^{5}\circ x^{4}=x^{6}\circ (x^{4})^{5}=x^{6}\circ x^{20}=(x^{20})^{6}=x^{120}$;

$\circ _{k=1}^{2}[-kx]=[-(2)x]\circ [-(2-1)x]=(-2x)\circ (-x)=-2(-x)=2x$;

$\circ _{k=-1}^{0}[x^{k}]=x^{(0)}\circ x^{(0-1)}=1\circ {x}^{(-1)}=1$;

$\circ _{k=1}^{5}[(3x+1)!]=(3x+1)!$

Current Applications

The composite function series can be used to define numbers of immense size in series form, typically those that are significantly larger than a googolplex. In particular, the series has been applied to express numbers used throughout Ramsey theory.

For example, Hulse uses the composite function series to convert Graham's function, and ultimately Graham's number, from recursive form to series form:

Graham's function is traditionally defined in recursive form using Knuth's up-arrow notation:

$g_{n}=\left\{{\begin{matrix}3\uparrow \uparrow \uparrow \uparrow 3,&n=1\\3\uparrow ^{g_{n-1}}3,&n\geq 2,n\in \mathbb {N} \end{matrix}}\right.$

In the first step, Knuth's up-arrow notation is converted to hyperoperation notation:

$a\uparrow ^{x}b=H_{x+2}(a,b)$

Then, the first few terms of Graham's function are expressed in hyperoperation notation:

$g_{1}=3\uparrow \uparrow \uparrow \uparrow 3=3\uparrow ^{4}3=H_{4+2}(3,3)=H_{6}(3,3)$

$g_{2}=3\uparrow ^{g_{2-1}}3=3\uparrow ^{g_{1}}3=3\uparrow ^{H_{6}(3,3)}3=H_{[H_{6}(3,3)+2]}(3,3)$

$g_{3}=3\uparrow ^{g_{3-1}}3=3\uparrow ^{g_{2}}3=3\uparrow ^{H_{[H_{6}(3,3)+2]}(3,3)}3=H_{[H_{[H_{6}(3,3)+2]}(3,3)+2]}(3,3)$

$g_{4}=3\uparrow ^{g_{4-1}}3=3\uparrow ^{g_{3}}3=3\uparrow ^{H_{[H_{[H_{6}(3,3)+2]}(3,3)+2]}(3,3)}3=H_{[H_{[H_{[H_{6}(3,3)+2]}(3,3)+2]}(3,3)+2]}(3,3)$

Next, Graham's function ($g_{n}$) is expressed in hyperoperation notation:

$g_{n}=H_{[H_{\ldots _{[H_{[H_{6}(3,3)+2]}(3,3)+2]}}(3,3)+2]}(3,3)$

Using the newly expressed definition of Graham's function, a composite function series is formed:

$[H_{x}(3,3)+0]\circ [H_{x}(3,3)+2]\circ [H_{x}(3,3)+2]\circ \ldots \circ [H_{6}(3,3)+2]$

The next step is to find an expression that can be used to represent the entire series. It is easy to start off by using the composite function series to express one value of Graham's function.

In the case below, the newly formed composite function series is $g_{64}$, or Graham's number, where the series consists of $64$ terms:

$g_{64}=[H_{x}(3,3)+0]\circ [H_{x}(3,3)+2]\circ [H_{x}(3,3)+2]\circ \ldots \circ [H_{6}(3,3)+2]$

The case is used to create two new sequences. The first sequence consists of the orders of the hyperoperations, and the second sequence consists of the values that are being added to each term. Both of these sequences are defined as the following:

$S_{1}=\{x,x,x,x,\ldots ,6\}$ (63 x's and one 6)

$S_{2}=\{0,2,2,2,\ldots ,2\}$ (one 0 and 63 2's)

The Kronecker delta,

$\delta _{i,j}={\begin{cases}0{\text{ if }}i\neq j,\\1{\text{ if }}i=j.\end{cases}}$

is then used to create a formula expressing each sequence:

$S_{1}=f_{1}(k)=(6-x)\cdot \delta _{k,1}+x$

$S_{2}=f_{2}(k)=2(1-\delta _{k,64})$

To express the sequences for each value of $n$ for $g_{n}$, replace $64$ with $n$:

$S_{1}=f_{1}(k)=(6-x)\cdot \delta _{k,1}+x$

$S_{2}=f_{2}(k)=2(1-\delta _{k,n})$

Now, the following expression can be used to represent Graham's function:

$\circ _{k=1}^{n}[H_{f_{1}(k)}(3,3)+f_{2}(k)]$

Graham's function and number as a composite function series

Therefore, the conversion is complete. Graham's function can be defined as the following using the composite function series:

Let $x,n\in \mathbb {N}$ with $n\geq x$. Then Graham's function ($g_{n}$) can be expressed as follows:

$g_{n}=\circ _{k=1}^{n}[H_{(6-x)\cdot \delta _{k,1}+x}(3,3)+2(1-\delta _{k,n})]$

Using the composite function series definition for Graham's function, Graham's number can also be expressed in series form:

$g_{64}=\circ _{k=1}^{64}[H_{(6-x)\cdot \delta _{k,1}+x}(3,3)+2(1-\delta _{k,64})]$