Let $f(x)=1/x$ and prove that $f[x_0,x_1,...,x_n]=\prod_{i=0}^nx_i^{-1}$.

Question

Let $f(x)=1/x$ and prove that $f[x_0,x_1,...,x_n]=\prod_{i=0}^nx_i^{-1}$. I'm sure how to approach this or even how/why we need $f(x)=1/x$. Any solutions or hints are greatly appreciated.

Answer 1

Assume for induction that the backward divided difference of $f$ satisfy

$$f[x_0,x_1,\ldots,x_k] = \frac{1}{x_0x_1\ldots x_k}$$

for $k=n$. We then find that

$$f[x_0,x_1,\ldots,x_n,x_{n+1}] \equiv \frac{f[x_0,x_1,\ldots,x_n]-f[x_1,\ldots,x_n,x_{n+1}]}{x_{n+1}-x_0} \\= \frac{\frac{1}{x_0x_1\ldots x_n} - \frac{1}{x_1\ldots x_nx_{n+1}} }{x_{n+1}-x_0} = \frac{1}{x_0x_1\ldots x_nx_{n+1}}$$

so the statement also holds for $k=n+1$. Since $f[x_0] = f(x_0) = \frac{1}{x_0}$ the statement holds for all $k$.