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The $\Gamma$ function is the analytic continuation of the factorial function. Is there a similar analog for multifactorials?

I am particularly interested in the double factorial. All Google has given me is the following formula relating the $\Gamma$ function to the double factorial for half integer values: $$\Gamma\left(n+\frac{1}{2}\right)=\frac{(2n-1)!!}{2^n}\sqrt{\pi}$$ But I want to double factorial nonintegers, so this is not really helpful.

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The Wikipedia article on "factorial" has some answers:… – Brad Feb 1 '13 at 4:20
"The Γ function is the analytic continuation of the factorial function." This makes it sound as if $\Gamma$ is the unique analytic function that interpolates factorial, but this is far from being the case. What criteria are you using to single out $\Gamma$ from all possible analytic interpolants? (Or maybe you mean something different.) – Andrés E. Caicedo Feb 1 '13 at 6:20
up vote 9 down vote accepted

Why is this not helpful? If you write the identity as $$(2n-1)!! = \frac{\Gamma(n + \frac{1}{2}) 2^n}{\sqrt{\pi}}$$ and then let $n = (x+1)/2$ you get $$x!! = \frac{\Gamma(\frac{x+2}{2}) 2^{(x+1)/2}}{\sqrt{\pi}}$$ The right side is now defined for any complex $x$ as long as the argument for the Gamma function is defined.

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Oh, of course. Thanks. This agrees with Brad's link also. – Alexander Gruber Feb 1 '13 at 6:32

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