Meaning of $f[x_1,\cdot\cdot\cdot,x_n]$

Question

I have a question in a problem set in numerical analysis. It states that for the function $f(x)=x^n$ and the $x_i$ points, show $f[x_1,\cdot\cdot\cdot,x_n]=\sum_i x_i$. What does this mean?

Answer 1

The divided difference of a function $f$ on the nodes $x_{1},\ldots,x_{n}$ is denoted by $f[x_{1},\ldots,x_{n}]$. Divided differences are defined inductively, by the rules $$ f[x_{i}]=f(x_{i})$$ and $$f[x_{i},\ldots,x_{i+j}]=\frac{f[x_{i+1},\ldots,x_{i+j}]-f[x_{i},\ldots,x_{i+j-1}]}{x_{i+j}-x_{i}}.$$