Is a factorial an algebraic function and an elementary function?

Question

Following is a question spun off from a comment I received:

is a factorial an elementary function and an algebraic function?

From elementary functions by Wikipedia

By starting with the field of rational functions, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.

So isn't a factorial a multiplication of finite polynomials, and therefore a polynomial, a rational function, an algebraic function, and an elementary function?

Added: Now I realized a factorial cannot be a polynomial, for that it doesn't make sense to talk about its degree while it does for a polynomial

Thanks!

Answer 1

To start basically factorial is really a function, and generally the Gamma function extends the factorial to the non-integer values.

To your best reference I have one beautiful article with me, let me suggest it , its here. The article is by Manjul Bhargava, it has the precise information what you are looking for.

Please read it and give your feed-back.

Edit : After thinking much on Mr.Srivatsan's comment , I came to a conclusion that $n!$ is not a polynomial, in fact $n!$ grows faster than $a^n$ for any $a$. Once you go out $a$ steps, adding $1$ to $n$ multiplies $a^n$ by $a$, while it multiplies $n!$ by $a$ by (at least) $a+1$.

And to add some interesting points,

1. The falling factorial $(x)_n$ is a binomial polynomial which is defined as $$(x)_n=x(x-1)....(x-(n-1))$$ for $n\ge 0$, and it can be related to the raising factorial $(x)^n$ ( defined as $(x)^n=x(x+1)...(x+n-1)$ ) as $$(x)_n=(-1)^n(-x)^{(n)}$$
2. The usual factorial ( that OP is talking about ) can be written as $$n!=(n)_n$$ which is not a polynomial anymore.

( Credits : Thanks Mr.Srivatsan for letting me know the difference ) .

Thank you,

Yours truly,

Iyengar.