Hyper-D notation

The Hyper-D and Hyper-E both consist of a sequence an of one or more positive integer arguments separated by hyperions (or hyper marks) #. We notate this as D[b](a1)#(a2)#...#(a(n)) and E[b](a1)#(a2)#...#(a(n)) in Hyper-D and Hyper-E notations, respectively. b is called the base — if it is omitted, as it often is, it defaults to 10. "D[b]d" and "E[b]d" are also both equal to "b^d".

The Hyper-D notation modifies multiple rules of the Saibian's Hyper-E notation as follows (differences are shown in purple):

• Rule 1. If there are no hyperions: D[b]x = E[b]x = b^x

  • For this rule, The Hyper-D and Hyper-E notations function the same with no hyperions.

• Rule 2. If the last entry is 0 (1 in Hyper-E):

  • Hyper-D: D[b](a1)#(a2)#(a3)#...#(a(n))#0 = D[b](a1)#(a2)#(a3)#...#(a(n))

  • Hyper-E: E[b](a1)#(a2)#(a3)#...#(a(n))#1 = E[b](a1)#(a2)#(a3)#...#(a(n))

• Rule 3. Otherwise (iteration last):

  • Hyper-D: D[b](a1)#(a2)#(a3)#...#(a(n-2))#(a(n-1))#(a(n)) = D[b]0#0#0#...#0#(D[b](a1)#(a2)#(a3)#...#(a(n-2))#(a(n-1))#(a(n) - 1))

  • Hyper-E: E[b](a1)#(a2)#(a3)#...#(a(n-2))#(a(n-1))#(a(n)) = E[b](a1)#(a2)#(a3)#...#(a(n-2))#(E[b](a1)#(a2)#(a3)#...#(a(n-2))#(a(n-1))#(a(n) - 1))

Extended Hyper-D notation

The Extended Hyper-D notation is similar to the Hyper-D and the Extended Hyper-E notations that allows multiple hyperions to appear between each entry. The number of hyperions following entry a(n) is represented by h(n). For the sake of this definition, #^n is a shorthand for n successive hyperion marks. This alteration for #^n only applies for the shorthand for the number of successive hyperion marks where the delimiter is just the copies of hyperion marks and is not accompanied by the delimiters from the later parts of the Extensible-E system, depending on the particular rules. For example, #^1 = ##, #^2 = ###, #^3 = ####, etc. On the other hand, #^#*#^1 = #^#*#, #^#^2 = #^##, #^#^#^3 = #^#^###, and so on. Saibian uses @ to indicate the rest of the expression such as Bowers uses # to indicate the rest of the array.

Here we show the alterations of the rules from the Extended Hyper-E notation:

• Rule 1. If there are no hyperions: D[b]x = E[b]x = b^x

  • For this rule, The Hyper-D and Hyper-E notations function the same with no hyperions.

• Rule 2. If the last entry is 0:

  • D[b]@#^[h(n-1)](a(n))#^[h(n)]0 = D[b]@#^[h(n-1)](a(n))

  • D[b]@#^[h(n-1)]0#^[h(n)]0 = D[b]@#^[h(n-1)]b

  • The modification of this rule also applies to the Cascading-D notation and further extensions as well.

• Rule 3. If h(n - 1) > 0:

  • D[b]@#^[h(n-2)](a(n-1))#^[h(n-1)]a(n) = D[b]@#^[h(n-2)]0#^[h(n-1) - 1]0#^[h(n-1) - 1]...#^[h(n-1) - 1]0#^[h(n-1) - 1](a(n-1))#^[h(n-1) - 1]b w/ a(n) - 1 0's separated by #^[h(n-1) - 1]. 

• Rule 4. Otherwise:

  • D[b]@#^[h(n-2)](a(n-1))#^[h(n)](a(n)) = D[b]@#^[h(n-2)](D[b]@#^[h(n-2)](a(n-1))#^[h(n)](a(n) - 1)) (note #^1 = ## and all the arguments in @ must be turned into 0). The modification of this rule also applies to the Cascading-D notation and further extensions as well.

Cascading-D notations and beyond

Further extensions of the Extensible-D System modifies certain rules of the existing Cascading-E notation, Extended Cascading-E notation, and the Hyper-Extended Cascading-E notation, as follows:

• In the Extended Cascading-D notation and the Hyper-Extended Cascading-D notation, A{n}B is equal to A^^^...^^^B with n+1 ^'s, while in the Extensible-E analog, A{n}B is equal to A^^^...^^^B with n ^'s.

• In the Extended Cascading-D notation and the Hyper-Extended Cascading-D notation, for n >= 1, A{x}#^#[n] is equal to A{x}###...### with n+1 #'s, while in the Extensible-E analog, A{x}#^#[n] is equal to A{x}###...### with n #'s. On the other hand, A{x}(#^#*#^#)[n] = A{x}(#^#*###...###) with n #'s after #^#, A{x}(#^#^#)[n] = A{x}(#^###...###) with n #'s after #^, and so on. This is because it will preserve the ordinal level increment.

• The remainder of the rules remains unchanged.