The hyperfactorial (Sloane and Plouffe 1995) is the function defined by
(1)
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(2)
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where is the K-function.
The hyperfactorial is implemented in the Wolfram Language as Hyperfactorial[n].
For integer values , 2, ... are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (OEIS A002109).
The hyperfactorial can also be generalized to complex numbers, as illustrated above.
The Barnes G-function and hyperfactorial satisfy the relation
(3)
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for all complex .
The hyperfactorial is given by the integral
(4)
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and the closed-form expression
(5)
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for , where is the Riemann zeta function, its derivative, is the Hurwitz zeta function, and
(6)
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also has a Stirling-like series
(7)
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has the special value
(8)
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(9)
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(10)
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where is the Euler-Mascheroni constant and is the Glaisher-Kinkelin constant.
The derivative is given by
(11)
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