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Ref: Euler's limit formula for the factorial function

I recently read about the Gamma function which works as a factorial operator for Natural Numbers. Most of the books even consider it as an extension of the factorial to Real Field.

So, my question is that is are there any other approaches to extend the Factorials to Reals or Positive Reals using Limits?

The most accurate I found was the one which Euler wrote to his first letter to Goldbach!

$$ s! = \lim \limits_{n\rightarrow\infty}\dfrac{n!}{(s+1)(s+2)(s+3)\dots(s+n)}(1+s)^n $$

Can the above be extended to Reals and decimals?

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  • $\begingroup$ There is essentially only one 'good' extension of the factorial; this is the Bohr–Mollerup theorem. "Good" here means that it obeys the same recurrence and initial conditions and is log-convex. $\endgroup$
    – Jair Taylor
    Commented Sep 6, 2019 at 23:11

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Indeed, the given form leads to one of the many different representations of the Gamma function, of which you can find other definitions on the Wikipedia page, all of which involve limits one way or the other using infinite series, products, or integrals.

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