Definitions #
Definition 1 (Geometric Function). A geometric function is formed whenever an initial value is continuously multiplied by a constant factor r.
$$ a_0 \xrightarrow{\times r} a_0r \xrightarrow{\times r} a_0r^2 \xrightarrow{\times r} a_0r^3 \xrightarrow{\times r} \cdots $$Definition 2 (Hypergeometric Function). Whenever a function’s term ratio depends on a rational function \(R(n)\), the function becomes hypergeometric. Geometric functions are a subcase of hypergeometric functions.
Since every term can be expressed as a product of consecutive ratios, the general term can be expressed as:
$$ a_{n}=a_{0}\prod_{j=0}^{n-1}R\left(j\right) $$Definition 3 (\(q\)-series). A “\(q\)-series” is defined as a series with summands of the form \((1-a)(1-aq) \dots(1-aq^{n-1})\), which often involves the products of the form \((1-aq^k)\).
Bases #
As a start, we can observe the initial dimensional growth in partition functions:
$$ \underbrace{\prod_{n=1}^{\infty} \frac{1}{1-q^n}}_{\mathcal{G}_1 \text{ (Geometric Line)}} \quad\xrightarrow{q}\quad \underbrace{\prod_{n=1}^{\infty} \left(\frac{1}{1-q^n}\right)^n}_{\mathcal{G}_2 \text{ (Plane)}} $$We will define the Geometric and Hypergeometric functions to be the base cases of the hierarchy, \(\mathcal{Y}_1\) and \(\mathcal{Y}_2\).
| Hierarchy Level | Function Type | Form |
|---|---|---|
| \(\mathcal{Y}_1\) | Geometric | \(a_{n}=a_{0}\left(r^{n-1}\right)\) |
| \(\mathcal{Y}_2\) | Hypergeometric | \(a_{n}=a_{0}\prod_{j=0}^{n-1}R\left(j\right)\) |
Function case comparison.
Ackermann Complexity #
We can connect our bases’ ratio generators to the Ackermann sequence for hyperoperations:
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\(A_2\) - Additive Ratio Growth \(\mathcal{Y}_1\)
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\(A_3\) - Multiplicative Ratio Growth \(\mathcal{Y}_2\)
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\(A_4\) - Exponential Ratio Growth \(\mathcal{Y}_3\)
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\(A_5\) - Tetration Ratio Growth \(\mathcal{Y}_4\)
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\(\vdots\)
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\(A_n\) - Growth \(\mathcal{Y}_{n-1}\)
The hyperoperation sequence forms a natural connection to the current hierarchy, as geometric functions grow according to a linear \(r\) encompassed by “additive ratio growth,” while hypergeometric ratios grow according to a factorial (multiplicative growth). To derive \(\mathcal{Y}_3,\) we search for a function family whose ratios grow exponentially: the \(q\)-series.
$$ (a;q)_n = \prod_{k=0}^{n-1} (1-aq^k) $$Operator #
We will define the operator \(\varphi\) to be the transform from \(\mathcal{Y}_n\) to the \(\mathcal{Y}_{n+1}\) Similarly, \(\varphi^{-1}\) will transform \(\mathcal{Y}_n\) to \(\mathcal{Y}_{n-1}\).
$$ \begin{matrix} \mathcal{Y}_n & \stackrel{\varphi}{\longrightarrow} & \mathcal{Y}_{n+1} \\ \uparrow^{\varphi^{-1}} & & \downarrow^{\varphi} \\ \mathcal{Y}_{n+1} & \stackrel{\varphi^{-1}}{\longleftarrow} & \mathcal{Y}_{n+2} \end{matrix} $$If we want to transform a function in our hierarchy, we can perform the following:
$$ \mathcal{Y}_{n+1} = a_0 \cdot \prod_{j=0}^{n-1}\varphi(R\left(j\right)) $$Combinatorics #
Theorem 4 (Bounded Box Partitions). MacMahon (1896) proved that the number of subset plane partitions of a bounded \(r\times s \times t\) box is given by:
$$ \prod_{i=1}^{r}\prod_{j=1}^{s}\frac{i+j+t-1}{i+j-1} $$We can express the formula in general hierarchy form using the following identity:
Identity 1 (Gamma Decomposition of a Product).
$$ \prod_{k=1}^{s}(x+k) = \frac{\Gamma(x+s+1)}{\Gamma(x+1)} $$We can group \(j\) outside of the sum for both the numerator and the denominator to apply the identity:
$$ \begin{aligned} & \prod_{i=1}^{r}\prod_{j=1}^{s}\frac{i+j+t-1}{i+j-1}\\ & = \prod_{i=1}^{r}\left(\prod_{j=1}^{s}\frac{\left(i+t-1\right)+j}{\left(i-1\right)+j}\right)\\ & = \prod_{i=1}^{r} \frac{\Gamma\left(i+t+s\right)\Gamma\left(i\right)}{\Gamma\left(i+t\right)\Gamma\left(i+s\right)} \end{aligned} $$Recalling our general hierarchy form, the inner product would simply be \(R(n)\). Since \(R(n)\) is composed of nested Gamma terms, \(N(r,s,t)\) must be in \(\mathcal{Y}_3.\) The formula that calculates partitions in 2D space is itself \(\mathcal{Y}_2\).
Existence #
\(\textbf{Open Question:}\) Does an operator/transform \(\phi\) exist, and how would it be defined?