Let $f \in C[a, b]$ and let $n+1$ distinct points $x_{0}, \ldots, x_{n} \in[a, b]$ be given.

(a) Define the divided difference $f\left[x_{0}, \ldots, x_{n}\right]$ of order $n$ in terms of interpolating polynomials. Prove that it is a symmetric function of the variables $x_{i}, i=0, \ldots, n$.

(b) Prove the recurrence relation

$$f\left[x_{0}, \ldots, x_{n}\right]=\frac{f\left[x_{1}, \ldots, x_{n}\right]-f\left[x_{0}, \ldots, x_{n-1}\right]}{x_{n}-x_{0}}$$

(c) Hence or otherwise deduce that, for any $i \neq j$, we have

$f\left[x_{0}, \ldots, x_{n}\right]=\frac{f\left[x_{0}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{n}\right]-f\left[x_{0}, \ldots, x_{j-1}, x_{j+1}, \ldots, x_{n}\right]}{x_{j}-x_{i}} .$

(d) From the formulas above, show that, for any $i=1, \ldots, n-1$,

$$f\left[x_{0}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{n}\right]=\gamma f\left[x_{0}, \ldots, x_{n-1}\right]+(1-\gamma) f\left[x_{1}, \ldots, x_{n}\right],$$

where $\gamma=\frac{x_{i}-x_{0}}{x_{n}-x_{0}}$.