# Divided difference coefficient of product of two functions

For any function $f$ and distinct reals $x_1,\ldots,x_n$, denote by $f[x_0,\ldots,x_n]$ the coefficient of $x^n$ of the minimal polynomial interpolating $f$ at $x_0,\ldots,x_n$.

Let $f$ and $g$ be real-valued functions defined on the real line and let $x_0,x_1,\ldots,x_n$ be $n+1$ distinct real numbers. Derive a formula for the $n$-th order divided difference $(fg)[x_0,x_1,\ldots,x_n]$ of the product function $fg$ at $x_0,x_1,\ldots,x_n$ in terms of the divided differences $f[x_0,x_1,\ldots,x_k]$ and $g[x_k,x_{k+1},\ldots,x_n]$ of $f$ and $g$, for $k=0,1,\ldots,n$.

For $n=1$, I've found that $(fg)[x_0,x_1]=f[x_0]g[x_0,x_1]+f[x_0,x_1]g[x_1]$. But I didn't find any nice form for $n=2$. If I can guess the correct formula, I'm quite sure it can be proved by induction.

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You'll want to look up the Leibniz formula. See this as well. – J. M. Apr 18 '12 at 16:14
Thank you very much! – matt Apr 19 '12 at 4:33

the answer is $\sum_{j=0}^{n}f[x_{0}, x_{1},..., x_{j}]g[x_{j}, x_{j+1},..., x_{n}]$