X-Sequence Hyper-Exponential Notation

X-Sequence Hyper-Exponential Notation is an extension of the hyper operators by SuperJedi224Almost Infinite, partly inspired by both Cascading-E notation and BEAF.
==Definition==
Main Definition
m{1}n = mn
m{α}1 = m
m{α+1}(n+1) = m{α}(m{α+1}n)
If α is a limit case: m{α}n = m{α[n]}m
Definition of α[n]
X[n] = n
(α+β)[n] = α+β[n]
(α*(β+1))[n] = α*β+α[n]
(α*β)[n] = α*β[n], if β is a limit case
α*1 = α
(αβ+1)[n] = αβ*α[n]
(αβ)[n] = αβ[n], if β is a limit case
α1 = α
(α↑↑X)[0] = 1
(α↑↑X)[n] = α(α↑↑X)[n-1]
α>(β+1) = (α>β)↑↑X
(α>β)[n] = α>β[n], if β is a limit case.
α>0 = α
(α>>X)[0]=0
(α>>X)[n+1]=α>(α>>X)[n]
==Growth Rate==
This notation is believed to have a limit ordinal of \(\zeta_0\) in the Fast-growing hierarchy.
==Examples==
4{3}7 = 4{2}4{2}4{2}4{2}4{2}4{2}4 (this is solved from right to left).
4{X+1}3 = 4{X}4{X}4 = 4{X}4{4}4 = 4{4{4}4}4.
4{X>X>X}3 = 4{X>X>3}4 = 4{X>((X↑↑X)↑↑X)↑↑X)}4
==In Other Notations==
a{c}b = \(a\uparrow^cb\)
a{X}b = \(a\uparrow^ba\)
a{X+1}b = {a,b,1,2}
a{X+2}b = {a,b,2,2}
a{X*2}b= {a,a,b,2}
a{X^2}b={a,a,a,b,2}
a{X^^2}b=a{X^X}b={a,b(1)2}
a{X^^3}b=a{X^X^X}b={a,b(0,1)2}
a{X^^4}b=a{X^X^X^X}b={a,b((1)1)2}
a{X^^X}b={a,b[1\2]2}
==Sources==
Category:Notations