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Denote $f(x)=x^n$ ($n$ is natural), I am trying to prove that $f[x_0,x_1,\ldots,x_n]=1$, $\{x_i\}$ are distinct $n+1$ real numbers.

I tried doing this by finding an interpolation of the function and by using Newton's form but what I did didn't lead to the solution.

I also tried using the definition as given in Wikipidia but that didn't help me either.

any ideas on how to prove this ?

[This question is taken from the book Numerical analysis by David Ronald Kincaid‏, Elliott Ward Cheney‏]

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Since $f[x_0,\dots,x_n]$ is constant, the statement follows from equality $$\lim_{(x_0,\dots,x_n)\to(\xi,\dots,\xi)} f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}$$ in Wikipedia. – Pierre-Yves Gaillard Dec 31 '11 at 14:54
You should probably turn this into an answer, Pierre. This is the key to OP's problem. – Patrick Da Silva Dec 31 '11 at 15:03
Dear @Patrick: Thanks!!! [I saw your comment by chance. It's safer (if you think of it) to use the @sign to ping someone.] – Pierre-Yves Gaillard Dec 31 '11 at 15:27
I know how to ping : sorry I didn't do it. – Patrick Da Silva Dec 31 '11 at 15:32
Dear @Patrick: Please, don't be sorry. The main point of my comment was to thank you. – Pierre-Yves Gaillard Dec 31 '11 at 15:53
up vote 5 down vote accepted

As kindly suggested by Patrick Da Silva, I'm turning my comment into an answer.

Let $x_0,x_1,\dots$ be distinct real numbers, let $f$ be a polynomial function on $\mathbb R$, and define $f[x_0,\dots,x_j]$ for $j=0,1,\dots$ recursively by $$ f[x_0]:=f(x_0), $$ $$ f[x_0,\dots,x_j]:=\frac{f[x_1,\dots,x_j]-f[x_0,\dots,x_{j-1}]}{x_j-x_0}\quad,\quad j\ge1. $$

It is straightforward to check that $f[x_0,\dots,x_j]$ is a homogeneous polynomial of degree $d-j$ in $x_0,\dots,x_j$ if $f$ is homogeneous of degree $d$.

In your case, $f[x_0,\dots,x_n]$ is constant, and the statement follows from equality $$ \lim_{(x_0,\dots,x_n)\to(\xi,\dots,\xi)} f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!} $$ in Wikipedia.

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thank you. I don't understand the need to claim : "f[x0,…,xj] is a homogeneous polynomial of degree d−j in x0,…,xj if f is homogeneous of degree d. " I also don't understand the need of the use of limit...from what I see it is sufficient to use the claim in… + the fact that the n-th derivative of x^n is n! and thus the interpolation of any given n points is n!. am I right ? thank you again for your help! – Belgi Dec 31 '11 at 15:54
Dear @Belgi: I think you're right. I didn't know this mean value theorem for divided differences. Your argument looks better than mine. You should post it as an answer to your own question, and eventually accept it unless there is a better answer. - I see that you've just accepted my answer. Thanks!!! In believe you can un-accept it if you want... Please feel free to do so! – Pierre-Yves Gaillard Dec 31 '11 at 16:06

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