Hyperfactorial

The hyperfactorial of a positive integer $n$ is the product of the numbers $1^1,\; 2^2,\; \backslash dots,\; n^n$. That is,{{r|oeis|summability}}

$$$$

H(n) = 1^1\cdot 2^2\cdot \cdots n^n = \prod_{i=1}^{n} i^i = n^n H(n-1).

Following the usual convention for the empty product, the hyperfactorial of 0 is 1. The sequence of hyperfactorials, beginning with $H(0)=1$, is:{{r|oeis}}

{{bi|left=1.6|1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... {{OEIS|A002109}}}}

The hyperfactorials were studied beginning in the 19th century by Hermann Kinkelin{{r|kinkelin|wilson}} and James Whitbread Lee Glaisher.{{r|glaisher|wilson}} As Kinkelin showed, just as the factorials can be continuously interpolated by the gamma function, the hyperfactorials can be continuously interpolated by the K-function.{{r|kinkelin}}

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials:

$$H(n)\; =\; An^\{(6n^2+6n+1)/12\}e^\{-n^2/4\}\backslash left(1+\backslash frac\{1\}\{720n^2\}-\backslash frac\{1433\}\{7257600n^4\}+\backslash cdots\backslash right)\backslash !,$$

where $A\backslash approx\; 1.28243$ is the Glaisher–Kinkelin constant.{{r|summability|glaisher}}

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when $p$ is an odd prime number

$$H(p-1)\backslash equiv(-1)^\{(p-1)/2\}(p-1)!!\backslash pmod\{p\},$$

where $!!$ is the notation for the double factorial.{{r|wilson}}

The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.{{r|oeis}}

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{{cite OEIS|1=A002109|2=Hyperfactorials: Product_{k = 1..n} k^k|mode=cs2}}

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{{citation

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• {{MathWorld|id=Hyperfactorial|title=Hyperfactorial|mode=cs2}}

Category:Integer sequences

Category:Factorial and binomial topics