Having seen some iterated factorial functions, let's see now a whole Hyperfactorial hierarchy of functions, based on the Ackermann hierarchy of operators:
| Function Name | Symbol | Natural Definition | Product Definition | Ackermann Function | Order | Analytic Continuation |
| Factorial | n! | 1*2*3*...*n | ∏k=1nk | N/A | N/A | Γ-Function |
| Superfactorial | G(n) | 1!*2!*3!*...*n! | ∏k=1nk! | N/A | N/A | Barnes G-Function |
| Double Factorial | H1(n)=n!! | (1+1)*(2+2)*(3+3)*...*(n+n) | ∏k=1n2*k | ∏k=1nA(1,k,k) | 1 | Γ((m+1)/2)*2(m+1)/2/sqrt(π) |
| Hyperfactorial | H2(n)=(n!)2 | 12*22*32*...*n2 | ∏k=1nk2 | ∏k=1nA(2,k,k) | 2 | Γ-Function |
| Hyperfactorial | H3(n) | 11*22*33*...*nn | ∏k=1nkk | ∏k=1nA(3,k,k) | 3 | K-Function |
| Hyperfactorial | H4(n) | 11*22*33*...*nn | ∏k=1nkk | ∏k=1nA(4,k,k) | 4 | ? |
| ... | ... | ... | ... | ... | ... | ... |
| Hyperfactorial | Hm(n) | 1^(m-2)1*2^(m-2)2*3^(m-2)3*...*n^(m-2)n | ∏k=1nk^(m-2)k | ∏k=1nA(m,k,k) | m | ? |
Can you order G(n) relative to the rest of the function hierarchy?