Hyperfactorial curiosity

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I've been familiarizing myself with the hyperfactorial, and I'm simply curious if it has an extension/analogue into the world of rational numbers, irrational numbers, and complex numbers like the common factorial has with the Gamma Function.

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If you're talking about the Hyperfactorial given at MathWorld, then yes. That page has details, though I don't want to make any quotes, since I'm not familiar enough with the area to make judgement calls about where to start/end a quote.