Aarex's Generalized Up-arrow Notation

Well, here you go. Here is Aarex's Generalized Up-arrow Notation.

Difficulty 1

• a{b}0 = a
• a{b}c (if 0 < c < 1) = a{b-1}((a^c-1)/(a-1))
• a{b}c = a{b-1}a{b}c-1

Difficulty 2

• a{b}0 = a
• a^^c (if 0 < c < 1) = a^((a^c-1)/(a-1))
• a{b}c (if 0 < c < 1) = a{b-1}((a{b-2}c-1)/(a-1))
• a{b}c = a{b-1}a{b}c-1

Difficulty 3

• a{b}0 = a
• a^^c (if 0 < c < 1) = a^((a^sin(c*pi/2)-1)/(a-1))
• a{b}c (if 0 < c < 1) = a{b-1}((a{b-2}sin(c*pi/2)-1)/(a-1))
• a{b}c = a{b-1}a{b}c-1