An Infinite Hierarchy of Hyperfactorial Functions

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 -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 ((m+1)/2)*2(m+1)/2/sqrt()
Hyperfactorial H2(n)=(n!)2 12*22*32*...*n2 k=1nk2 k=1nA(2,k,k) 2 -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?

Back to Mathematics

Web Analytics Made Easy -
StatCounter

Valid HTML 4.01 Transitional