User blog:Dhacorrea/Greek factorials: an extension to superfactorials

Thinking about superfactorials, especially related to that defined by Pickover, I thought why not try an extension to such functions? Pickover superfactorial is defined as \(n\$=\underbrace{n!^{n!^{n!^{.^{.^{.^{.}}}}}}}_{n!}\) that is the same of \(n![4]n!\). The idea of Greek factorials is to be an extension of the hyperoperation, exponent and base number of the superfactorial function. Initially I suggest eight Greek factorials as below: Greek factorials functions grows tremendously faster than superfactorial. The "slowest" one, \(n\alpha\$\), is much faster than Pickover superfactorial, see the example of \(3\$\) and \(3\alpha\$\) below:
 * \(n\alpha\$ = n![n+2]n! = n!\underbrace{\uparrow...\uparrow}_{n}n!\);
 * \(n\beta\$ = n![n!+2]n! = n!\underbrace{\uparrow...\uparrow}_{n!}n!\);
 * \(n\gamma\$ = n\$[n]n! = n\$\underbrace{\uparrow...\uparrow}_{n}n!\);
 * \(n\delta\$ = n![n]n\$ = n!\underbrace{\uparrow...\uparrow}_{n}n\$\);
 * \(n\theta\$ = n![n!]n\$ = n!\underbrace{\uparrow...\uparrow}_{n!}n\$\);
 * \(n\Sigma\$ = n\$[n]n\$ = n\$\underbrace{\uparrow...\uparrow}_{n}n\$\);
 * \(n\Phi\$ = n\$[n!]n\$ = n\$\underbrace{\uparrow...\uparrow}_{n!}n\$\);
 * \(n\Omega\$ = n\$[n\$]n\$ = n\$\underbrace{\uparrow...\uparrow}_{n\$}n\$\)

\(3\alpha\$ = \underbrace{6\uparrow6\uparrow...\uparrow6\uparrow6}_{\underbrace{6\uparrow6\uparrow...\uparrow6\uparrow6}_{\underbrace{6\uparrow6\uparrow...\uparrow6\uparrow6}_{\underbrace{6\uparrow6\uparrow6\uparrow...\uparrow6\uparrow6\uparrow6}_{\underbrace{6\uparrow6\uparrow6\uparrow6\uparrow6\uparrow6}_{6}}}}}\)