User blog comment:PsiCubed2/Formal definition of Letter Notation up to R for integer arguments/@comment-28606698-20170425102247

That is good notation. Unfortunately I missed it. I did not test but it looks like

Λ(α,b)=10↑(α)b

using the extension of up arrows notation to transfinite ordinals.

Indeed Kx={10,x,1,2}

I have only several remarks about fundamental sequences:

1) For β=ω^(β₁) (with β₁<β a limit ordinal) we have β[n] = ω^(β₁[n]) i.e. [n] missed in exponent

2) For β=φ(β₁,0) (with β₁<β a limit ordinal) we have β[n]=φ(β₁[n],0) i.e. [n] instead [n+1] in left part

3) For β=φ(β₁,α+1) (with β₁<β a limit ordinal) we have β[n]=φ(β₁[n],φ(β₁,α)+1)

4) For β=φ(α,β₁) (with β₁<β a limit ordinal) we have β[n]=φ(α,β₁[n])