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show if $\phi (x) = f(x)g(x)$, this is valid:
$\phi [x_0,x_1,...,x_n]=\sum\limits_{r=0}^n f[x_0,x_1,..,x_r]g[x_r,x_{r+1},...,x_n]$ by induction.

I have tried to prove it by the divided differences formula but things are standing still at the moment.

EDIT: I didn't understand this proof, but you should look at this as a reference; http://www.sosmath.com/CBB/viewtopic.php?t=31735

It is also known as Leibniz formula

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Can you clarify the notation? Are $\phi,f,g$ polynomials in one variable or in multiple variables? In other words how are $f(x)$ and $f[x_0,...,x_n]$ related? Also is there a difference between the two kinds of brackets you use? –  Simon Markett Mar 26 '13 at 12:15
    
Simon: $f\left[x_0,...,x_n\right]$ is a so-called divided difference of $f$; see en.wikipedia.org/wiki/Divided_differences#Notation . –  darij grinberg Mar 28 '13 at 1:12

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