The **Cascading-E notation** (E^ for short) is a further extension and generalization of Hyper-E notation introduced by Sbiis Saibian in January 22, 2013.^{[1]} The limit of its growth rate is comparable to \(f_{\varepsilon_0}(n)\) in FGH, making it about as powerful as tetrational arrays in BEAF.

E^ is part of the general Extensible-E System, which includes all future extensions of the notation, such as Extended Cascading-E Notation.

## Terminology Edit

### Hyper-product and cascaders Edit

Separators of the form \(\#^{X}*\#^{X}\cdots\#^{X}*\#^{X}\) are known as a **hyper-product of cascaders** and each \(\#^{X}\) is a single **cascader**. X can be any positive integer or itself can be a hyper-product of cascaders. This recursive definition defines all possible separators in Cascading-E. Consequently, even separators in the form \(\#^{n}\) where n is a positive integer are considered a hyper-product of cascaders.

### Key Band Edit

The **Key Band** is a cascader of the form \(\#^{n}\) (n is a positive integer), situated at the right most position of the hyper-product.

The following search-algorithm finds and defines the Key Band:

- Begin at ground level (level 0), and proceed to step 2.
- For the current level, find the last cascader of the current hyper-product, and proceed to step 3.
- If this cascader is in the form \(\#^{n}\), then it is the Key Band by definition, otherwise proceed to step 4.
- Go up to the next exponent level of the last cascader and go back to step 2.

## Definition Edit

Let Ea1&a2& ... &an be any expression in E^, where a1 through an are n positive integers, and all & are *hyper-products* (which may or may not be distinct.) Each individual & may be chosen from the set of legal *separators*.

Below are the 5 formal rules of E^. Let \(\&_k\) be the kth hyper-product and \(L(\&_k)\) be the last cascader of the kth hyper-product.

- Rule 1. With no hyperions, we have \(E(a)b = a^b\).
- Rule 2. If \(L(\&_{n-1}) \neq \#^n\) (the last cascader is not of the form \(\#^{n}\)):
- \(E(a)b@X\#^{(X\#^{n})}@c = E(a)b@X\#^{(X\#^{n-1_)^{b}}}@b\) (@ indicates the unchanged remainder of the expression.)

- Rule 3. If the last argument is 1, it can be removed: \(E@a\#^{n}1 = E@a\)
- Rule 4. \(L(\&_{n-1}) = \#^n\) and \(\&_k \neq \#\):
- \(E@aX\#^{n}b = E@aX\#^{n-1}aX\#^{n}(b-1)\).

- Rule 5. Otherwise:
- \(E@a\#b = E@(E@a\#(b-1))\)

## Examples Edit

- E10#^#3 = E10###10
- E10#^#10#^#3 = E10#^#10###10
- E10#^#*#3 = E10#^#10#^#10
- E10#^##3 = E10#^#*#^#*#^#10
- E10#^#^#3 = E10#^###10
- E100#^^#3 = E100#^#^#100

## Sources Edit

- ↑ Saibian, Sbiis. Cascading-E Notation. Retrieved 2013-05-02.