Question 169
Let $d$ be a fixed positive integer. Prove that there exists a unique polynomial $S(n)$ such that for every integer $n \geq 0$, $S(n) = \sum_{k=0}^{n} k^{d} = 0^{d} + 1^{d} + \cdots + n^{d}$. Also prove that $S(n)$ can be expressed in the form
$$
c_{0} + c_{1}(2n+1) + c_{2}(2n+2)^{2} + \cdots + c_{d+1}(2n+1)^{d+1}
$$
where the $c_{i}$ are rational numbers such that $c_{i} c_{i+1} = 0$ for $i = 0, 1, \ldots, d$.
$$
c_{0} + c_{1}(2n+1) + c_{2}(2n+2)^{2} + \cdots + c_{d+1}(2n+1)^{d+1}
$$
where the $c_{i}$ are rational numbers such that $c_{i} c_{i+1} = 0$ for $i = 0, 1, \ldots, d$.