Expand-E notation

The Expand-E notation is an extension of Hyper-Extended Cascading-E Notation made by Aarex.https://docs.google.com/document/d/1eZIXQucE2xev2rIortfDJEp9pwOkWgJW9RHWScnCIbM/edit#heading=h.rc168y50hyfk Retrieved Aug 28 2024
==Definition==
Copied from the only available source 
“This consults #xE^ with new changes as following:
*Any positive integer is a member of the successor.
*Let *(a) denotes {} set, which is a set of {b}^c, where b and c are hyperion sums or hyperproducts.
***(a) replaces the {n} set from #xE^.
**a*(b)c is a set of x^, where *(b) is a {} set.
***n ^s denotes {n}^1.
***{{...{{a}}...}} with b braces denotes {a}^b.
***{a,b,c,d} denotes a{b}^(c)d.
a*(b)c =
*I: For *(b) = ^ = {1}^1: Do E^ rules normally.
*II: For c = # and *(b) = {d}^e:
**II-a: For d is lim: a({d}^e)#[n] = a({d[n]}^e)#
**II-b: For d is succ:
***n = 1: a({d}^e)#[1] = a
***n > 1: a({d}^e)#[n] = a({d-1}^e)(a({d}^e)#[n-1])
**II-c: For d = 1 and e is lim:
***a({1}^e)#[n] = a({1}^e[n])#
**II-d: For d = 1 and e is succ:
***n = 1: a({1}^e)#[1] = a
***n > 1: a({1}^e)#[n] = a({a({1}^e)#[n-1]}^(e-1))a
*III: For c = p*#:
**n = 1: a*(b)(p*#)[1] = a*(b)p
**n > 1: a*(b)(p*#)[n] = a*(b)p>a*(b)(p*#)[n-1]
*IV: For c is lim other than c = # and c = p*#:
**a*(b)c[n] = a*(b)(c[n])
When (p*#) can’t be applied by all existing rules, consult the hyperion sum. This is only applied by taking the fundamental sequence of a limit member.
*(p*#)[1] = p
*(p*#)[n] = p+(p*#)[n-1]
a = b + c, where b is a hyperproduct or hyperion sum and c is a hyperproduct or a positive integer:
*c is 0: a = b
*c is lim:
**a is a member of lim.
**(b+c)[n] = b+(c[n])
*c is succ:
**a is a member of succ.
**(b+c)-1 = b+(c-1)
Also consult the fundamental sequence of #: #[n] = n”.
==Sources==
Category:Notations
Category:Functions
Category:Numbers by Aarex Tiaokhiao