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?

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!

From my perspective your question may be good, but its not phrased properly, I can't understand your main doubt, may be the question can be reformulated more neatly , but is this problem only for me ?, if so sorry !! @Tim — IDOK, Jan 12 '12 at 15:45
@Sasha: $n!=n \times (n-1) \times \cdots \times 1$. I am not claiming it is true. I am not sure, but it seems like a multiplication of finite polynomials. — Tim, Jan 12 '12 at 15:50
"...in 1887 Holder proved that the gamma function is transcendentally transcendental, which incidentally gives a naturally occurring example of an infinitely differentiable function that is far more non-elementary than (the usual transcendental functions)..." — J. M. ain't a mathematician, Jan 12 '12 at 15:55
@J.M.: Thanks! What is that "naturally occurring example of an infinitely differentiable function that is far more non-elementary than (the usual transcendental functions)"? Are you saying it is the factorial? — Tim, Jan 12 '12 at 16:08
The gamma function (factorial), of course. I quoted and linked to Dave's answer; you might wish to peruse the references he gave. — J. M. ain't a mathematician, Jan 12 '12 at 16:09
@Tim : your idea is very beautiful, but I need to tell sorry as sometimes humans assert something, so I never like such assertions , as I know that one man's sweet is another man's poison, If I don't understand , it doesn't mean that nobody understands it . — IDOK, Jan 12 '12 at 16:18
@Tim : Hey you took me in wrong sense, I was saying I asserted something , you understand ? , I wanted to say humans assert something, I was referring to myself ( I mean that I asserted that your question is not properly presented , but its only for me ), I never commented you, anyway lets remove these comments, I will remove my one after some time, Thanks a lot, and I apologize if I caused you any trouble — IDOK, Jan 12 '12 at 16:26
@Tim : I have edited your question, you are at liberty to roll-back at any time . — IDOK, Jan 12 '12 at 16:41

score 5 · Accepted Answer · edited Nov 06 '22 at 22:42

5

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,

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 $$(x)_n=(-1)^n(-x)^{(n)}$$
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.

edited Nov 06 '22 at 22:42

mrjink

103

answered Jan 12 '12 at 16:23

IDOK

5,300

4

Your first sentence is incorrect: the factorial function is not a polynomial. [In fact, it grows faster than any polynomial.] I think you just assumed what the OP claimed to be true. Added note: I did not downvote. – Srivatsan Jan 12 '12 at 16:37
@Srivatsan : I am not an expert to argue further, but a polynomial in sense I mean a function of a single variable and its a function by definition as its a product that gives an $n$-th degree polynomial of single variable . – IDOK Jan 12 '12 at 16:44
3

Yes, it is of course a function of $n$ - no doubt. But it is not a polynomial: because the number of terms in the product grows unbounded. – Srivatsan Jan 12 '12 at 16:49
Ok thank you sir, I have fixed it, and I came to know about the difference , Thanks a lot !! @Srivatsan – IDOK Jan 12 '12 at 16:52
@Srivatsan: Why is "the number of terms in the product grows unbounded" not for a polynomial? – Tim Jan 12 '12 at 16:53
@Tim : but have you seen the article of Manjul ? – IDOK Jan 12 '12 at 16:58
@iyengar: Thanks! I haven't completed it yet. – Tim Jan 12 '12 at 17:00
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@Tim: A polynomial has a fixed degree, independent of its argument ($n$ in this case). The product $n(n-1)(n-2) \cdots 1$ contains $n$ terms, which is not allowed. – Srivatsan Jan 12 '12 at 21:37
@iyengar "Prof" and "sir" have specific meanings; they are not titles we can add to people's names at will. It is best to avoid using such (unnecessary, in my opinion) qualifiers in a professional forum such as this. I can understand that you only intend to display respect, but it only comes off to the rest of community as an abuse of the title. I am not a professor; so please remove it from my name. Thank you and regards, – Srivatsan Jan 13 '12 at 16:46
@Srivatsan : I have fixed it sir. – IDOK Jan 13 '12 at 17:13
@Iyengar The link to the "beautiful" article is broken :( – SAH Nov 02 '16 at 00:06

Is a factorial an algebraic function and an elementary function?

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