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One usually introduces the Gamma function to define a derivative of the factorial. However couldn't one define a derivative over integers like $$ f'(n) = \frac{f(n+1) - f(n)}{1} ? $$ Such a discrete alternative to the usual derivative would then allow us to simply compute a derivative of the factorial like $$ (n!)' = (n+1)! - n! = (n+1)n!-n! = n\ n! $$ My question is now: Is this the correct way to define a derivative over integer numbers and is the result for the derivative of the factorial correct? And how would one then compute the discrete integral/sum for the factorial?

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    $\begingroup$ Knuth's Concrete Mathematics has a lot about this difference operator. $\endgroup$ – Matthew Leingang Feb 17 '15 at 21:29
  • $\begingroup$ If you are interested in other ways to calculating the derivative of the factorial, see here: math.stackexchange.com/questions/1633014/… Another note is that taking the derivative of Stirling's approximations might be a good approach, if we are to avoid the Gamma function. $\endgroup$ – Simply Beautiful Art Jun 30 '16 at 16:07
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It's perfectly reasonable. These are called forward-finite-difference operators. On the same page of the link, you'll find a section on calculus for finite difference operators.

On the other hand, you might be interested in the definition of the Gamma function, which generalizes the factorial to non-integers and has a derivative in the classical sense.

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