X-Sequence Hyper-Exponential Notation

X-Sequence Hyper-Exponential Notation is an extension of the hyper operators by SuperJedi224, 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

Growth Rate
This notation is believed to have a limit ordinal of ε0 in the Fast-growing hierarchy

Examples
4{3}5=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{n}4, where n=4{4}4.

In Other Notations
a{c}b=a↑cb

a{X}b=a↑ba

a{X+1}b={a,b,1,2}

a{X*2}b={a,a,b,2}