Other representations of factorial

Question

I have a little question: which other representation of factorial $n!$ without using the factorial?

Is there any definition of factorial as a series? or any other way?

Thanks in advanced.

Answer 1

It depends on what you mean by "without using the factorial". The typical definition is usually $$ n!=\prod_{k=1}^n k $$ meaning multiply all the natural numbers less than or equal to $n$ together. It can also be defined recursively (or as a sequence) where $f_0=1$ and $f_n=n\cdot f_{n-1}$.

Others have mentioned the Gamma function, which is a generalized factorial defined by an integral.

It can also be represented as an $n$th derivative, where $$ n!=\frac{d^n}{dx^n}x^n $$ which reduces to a natural number for natural $n$ due to the power rule.

I don't know of any definition of a factorial as a series, however, it is a part of some useful ones. For example, $$ \sum_{x=0}^{\infty}\frac{1}{x!}=e $$ You could define factorial some other ways, like for $n\ge1$, you could say $n!=n!!(n-1)!!$, but that just makes things more complicated without really giving any benefit.

I'm not sure why you need or want another definition of factorial, but there should be something here to help you.

Answer 2

The Gamma function $\Gamma(t)=\int_0^\infty x^{t-1}e^{-x}\;dx$ extends the factorial to complex arguments. For natural numbers $\Gamma(n+1)=n!$

Answer 3

There's a very useful analytic generalization of $n!$: $$x!=\int_0^\infty t^x e^{-t}\, dt$$ This converges on $(-1,\infty)$ and is closely related to the Gamma function, which is a shifted version of this (so that it converges on $(0,\infty)$.

It can even be (uniquely) analytically extended to the whole complex plane except poles at $-1, -2, \ldots$