New mathematical object - hyperanalytic function

[Daniel]

How is this related to tetration?

Directly.
For example, function
\( \mathbb{R}(x)=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x-nL}{\sigma})^{2}} \)

has approximation

\( A\left( x \right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}(1+2\alpha cos(2\pi x ) \)

{Please fix, doesn't parse}
\( +2\sum_{i=1}^{\infty} \alpha^{4^i}( \cos ( 2i \times 2\pi x ) -1 )
+\frac{2}{\mathbb{W}_{max}}\sum_{i=1}^{\infty}\alpha^{9{i}^2}\left( cos\left(3 \times (2i-1)\times 2\pi x \right)
-cos\left( (2i-1) \times 2\pi  x \right) \right) \),

where

\( \alpha\left(\sigma\right)=\frac{1}{2}\frac{\mathbb{R}_{max}-\mathbb{R}_{min}}{\mathbb{R}_{max}+\mathbb{R}_{min}} \).