Inmathematics, and more specificallynumber theory, thehyperfactorial of a positiveinteger${\displaystyle n}$ is the product of the numbers of the form${\displaystyle x^x}$ from${\displaystyle 1^1}$ to${\displaystyle n^n}$.

Thehyperfactorial of a positive integer${\displaystyle n}$ is the product of the numbers${\displaystyle 1^1,2^2,\dots ,n^n}$. That is,^[1][2]$${\displaystyle H(n)=1^1\cdot 2^2\cdot \cdots n^n=\prod _{i=1}^n{i}^i=n^{n}H(n-1).}$$Following the usual convention for theempty product, the hyperfactorial of 0 is 1. Thesequence of hyperfactorials, beginning with${\displaystyle H(0)=1}$, is:^[1]

The hyperfactorials were studied beginning in the 19th century byHermann Kinkelin^[3][4] andJames Whitbread Lee Glaisher.^[5][4] As Kinkelin showed, just as thefactorials can becontinuously interpolated by thegamma function, the hyperfactorials can be continuously interpolated by theK-function.^[3]

Glaisher provided anasymptotic formula for the hyperfactorials, analogous toStirling's formula for the factorials:$${\displaystyle H(n)=A{n}^{(6n^2+6n+1)/12}{e}^{-n^2/4}\left(1+\frac{1}{720n^2}-\frac{1433}{7257600n^4}+\cdots \right),}$$where${\displaystyle A\approx 1.28243}$ is theGlaisher–Kinkelin constant.^[2][5]

According to an analogue ofWilson's theorem on the behavior of factorialsmoduloprime numbers, when${\displaystyle p}$ is anodd prime number$${\displaystyle H(p-1)\equiv (-1{)}^{(p-1)/2}(p-1)!!(\mathrm{mod}p),}$$where${\displaystyle !!}$ is the notation for thedouble factorial.^[4]

The hyperfactorials give the sequence ofdiscriminants ofHermite polynomials in their probabilistic formulation.^[1]

• Weisstein, Eric W.,"Hyperfactorial",MathWorld

• Integer sequences
• Factorial and binomial topics

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