Polynomials $P_{r}(X)$ for $r \geq 0$ are defined by

$$\begin{aligned} &P_{0}(X)=1 \\ &P_{r}(X)=\frac{X(X-1) \cdots(X-r+1)}{r !}=\prod_{i=1}^{r} \frac{X-i+1}{i} \quad \text { for } r \geq 1 \end{aligned}$$

Show that $P_{r}(n) \in \mathbb{Z}$ for every $n \in \mathbb{Z}$, and that if $r \geq 1$ then $P_{r}(X)-P_{r}(X-1)=$ $P_{r-1}(X-1)$.

Prove that if $F$ is any polynomial of degree $d$ with rational coefficients, then there are unique rational numbers $c_{r}(F)(0 \leq r \leq d)$ for which

$$F(X)=\sum_{r=0}^{d} c_{r}(F) P_{r}(X)$$

Let $\Delta F(X)=F(X+1)-F(X)$. Show that

$$\Delta F(X)=\sum_{r=0}^{d-1} c_{r+1}(F) P_{r}(X)$$

Show also that, if $F$ and $G$ are polynomials such that $\Delta F=\Delta G$, then $F-G$ is a constant.

By induction on the degree of $F$, or otherwise, show that if $F(n) \in \mathbb{Z}$ for every $n \in \mathbb{Z}$, then $c_{r}(F) \in \mathbb{Z}$ for all $r$.