implement hyperfactorial/K-function and superfactorial/G-function

[Patashu]

http://mrob.com/pub/math/hyper4.html#real-hyper4

"There are three other functions that have been extended to the reals in ways that seem promising: the factorials by the Gamma function, hyperfactorials by the K-function, and the lower (Simon and Plouffe 1995) superfactorial by Barnes' G-function."

http://mrob.com/pub/math/numbers-9.html#hyperfactorial

The hyperfactorials are: 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, ... (Sloane's A2109). The hyperfactorials can be extended to the real numbers, the result is the K-function, which is related to Barnes' G-function, the Gamma function and the Riemann Zeta function. n hyperfactorial is equivalent to K(n+1).

Hyperfactorials: Product_{k = 1..n} k^k.

http://mrob.com/pub/math/numbers-11.html#superf1

288 = 4!×3!×2!×1! = 44+33+22+11

(4 superfactorial by the Sloane-Plouffe definition)

This is the value of "4 superfactorial" by the lower (Sloane and Plouffe 1995) definition of "superfactorial": 4!×3!×2!×1! = 24×6×2 = 288. By a rather nifty coincidence, it is also equal to 44+33+22+11 = 256+27+8+1. See also 34560, 5056584744960000, and 2.703176857×10^6940.

Barnes' G-function Barnes' G-function is to superfactorials as the

Gamma function is to normal factorials. Barnes' G-function can be (very slowly) calculated by the formula:

G(z) = 2πz/2 e-[z(z+1)+γ z2]/2 PRODUCT(n=1..inf)[(1+z/n)n e-z+z2/(2n)]

where γ is the Euler-Mascheroni constant. For sufficiently large values of z you can use the approximation:

G(n) ≈ (e1/12/A) nn2/2-1/12(2π)n/2e-3n2/4

where A is the Glaisher-Kinkelin constant. See the MathWorld page 86 for more.

another page with definitions of both:

http://mrob.com/pub/math/largenum-3.html