Hyperfactorial

The hyperfactorial $H(n) = \prod_{k=1}^n k^k$ , continued to complex arguments: $H(z) = \Gamma(z+1)^z / G(z+1)$ .

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageHyperfactorial!

Hyperfactorial(n)the hyperfactorial of n.

Typed signatures

Type	Package	Applies
`(number) -> number`	`analytic`	always

Domain: Special functions

Details

$H(0) = 1$ , the empty product.
Continued off the nonnegative integers by $H(z) = \Gamma(z+1)^z / G(z+1)$ (G = BarnesG); numeric there, checked against a Wolfram kernel to double precision.
Exact at a nonnegative integer -- an arbitrarily large exact product, not a decimal.
Real, nonnegative domain only: BarnesG's zeros at the nonpositive integers give the continuation poles there, and no reference example calls for a negative or complex argument.

Examples

3 more kept as data, checked against the oracles

Basic #

$1^1\,2^2\,3^3\,4^4$

\mathrm{Hyperfactorial}(4)

Out

27\,648

The empty product

\mathrm{Hyperfactorial}(0)

Out

1

Listable

\mathrm{Hyperfactorial}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack)

Out

\bigl\lbrack1, 4, 108, 27\,648, 86\,400\,000, 4\,031\,078\,400\,000\bigr\rbrack

Continued off the integers

\mathrm{Hyperfactorial}(0.5)

Out

0.880\,449\,235\,173\,423\,4

$H(10)$ , well past double range — exact bigint arithmetic, not a rounded double.

\mathrm{Hyperfactorial}(10)

Out

215\,779\,412\,229\,418\,562\,091\,680\,268\,288\,000\,000\,000\,000\,000

Scope #

An exact non-integer argument stays symbolic — N() gives the float continuation

\mathrm{Hyperfactorial}(\frac{1}{2})

Out

\mathrm{Hyperfactorial}(\frac{1}{2})

About

Examples	9
Checked against	Wolfram Language 8 agree1 differ
Crosswalk	Wolfram Language

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/hyperfactorial.ts