Numbers and Sets | Part IA, 2004

Polynomials Pr(X)P_{r}(X) for r0r \geq 0 are defined by

P0(X)=1Pr(X)=X(X1)(Xr+1)r!=i=1rXi+1i for r1\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 Pr(n)ZP_{r}(n) \in \mathbb{Z} for every nZn \in \mathbb{Z}, and that if r1r \geq 1 then Pr(X)Pr(X1)=P_{r}(X)-P_{r}(X-1)= Pr1(X1)P_{r-1}(X-1).

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

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

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

ΔF(X)=r=0d1cr+1(F)Pr(X)\Delta F(X)=\sum_{r=0}^{d-1} c_{r+1}(F) P_{r}(X)

Show also that, if FF and GG are polynomials such that ΔF=ΔG\Delta F=\Delta G, then FGF-G is a constant.

By induction on the degree of FF, or otherwise, show that if F(n)ZF(n) \in \mathbb{Z} for every nZn \in \mathbb{Z}, then cr(F)Zc_{r}(F) \in \mathbb{Z} for all rr.

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