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We know that the gamma function is
$$\Gamma(n+1)=n!=\int_0^\infty e^{-x}x^n dx=I_n$$ It can be easily shown, via integration by parts, that $$I_n=nI_{n-1}$$ which confirms its suitability as a continuous function to represent the factorial function.

Question
Is it possible to construct another integral $J_n$ or continuous function $f_n$ that would have this property, i.e. $J_n=nJ_{n-1}$ or $f(n)=nf(n-1)$, or show that no such other integral or function exists?

Hypergeometricx
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    Quote from Wikipedia: "There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points. The gamma function is the most useful solution in practice, being analytic (except at the non-positive integers), and it can be defined in several equivalent ways. However, it is not the only analytic function which extends the factorial, as adding to it any analytic function which is zero on the positive integers, such as k sin mπx, will give another function with that property" – Clement Yung Jun 17 '20 at 16:39
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    If I remember correctly, the ${\Gamma}$ tends to be preferred since it's the only continuous extension that satisfies three different properties (log convexity being one of those?) I can't remember exactly though – Riemann'sPointyNose Jun 17 '20 at 16:56

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You could take $g(x) \Gamma(x)$ for any function $g$ that is periodic with period $1$.

Robert Israel
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