Superfactorials

Description

Superfactorials now have been requested multiple times. They are defined:

n$ = \prod_{i=1}^n i! = G(n+2)

A simple notation, that is common enough to warrant its own command. The G here is an analytical continuation, so unlike multitermials and subfactorials (both of which I cannot find one for) it can relatively easily support decimals. One consideration to take is efficiency, we will need to approximate way earlier.

There is a sterling-like approximation:

\ln G(1+z) ∼ z^2\left(\frac{1}{2}\ln z-\frac{3}{4}\right)+\frac{1}{2}\ln(2\pi)z-\frac{1}{12}\ln z+\zeta'(-1)+\sum_{k=1}^n\frac{B_{2k+2}}{4k(k+1)z^{2k}}+O\left(\frac{1}{z^{2n+2}}\right)

Where B_k is the k-th Bernulli number approximated by:

 B_{2n} ∼ (-1)^{n-1}4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}

If that proves too complex (and inefficient) there is a simpler form:

 \ln G(1+z) ∼ z^2\left(\frac{1}{2}\ln z-\frac{3}{4}\right)+\frac{1}{2}\ln(2\pi)z-\frac{1}{12}\ln z+\zeta'(-1)+O\left(\frac{1}{z}\right) 

Reasoning

This has been requested here:

• https://www.reddit.com/r/unexpectedfactorial/comments/1lo5e38/comment/n0kar1q
• https://www.reddit.com/r/unexpectedfactorial/comments/1ln30oj/petition_to_add_superfactorial/ (whole post)

Resources

• https://en.wikipedia.org/wiki/Barnes_G-function
• https://mathworld.wolfram.com/BarnesG-Function.html
• https://mathworld.wolfram.com/BernoulliNumber.html
• https://mathworld.wolfram.com/Superfactorial.html

No files, tests, or entry points are named. Start by reviewing the existing factorial command behavior and the Barnes G and superfactorial resources linked in the issue; define supported inputs, notation, approximation accuracy, and completion tests before implementation.