(i) Let $f ( x ) = a _ { n } x ^ { n } + \cdots + a _ { 1 } x + a _ { 0 }$ be a polynomial, where $a _ { 0 } , \ldots , a _ { n }$ are real numbers with $a _ { n } \neq 0$. Define the ``discrete derivative of $f$'', denoted $\Delta f$, to be the function given by $\Delta f ( x ) = f ( x ) - f ( x - 1 )$. Show that $\Delta f$ is also a polynomial and find its leading term.
(ii) For integers $n \geq 0$, define polynomials $p _ { n }$ of degree $n$ as follows: $p _ { 0 } ( x ) = 1$ and for $n > 0$, let $p _ { n } ( x ) = \frac { 1 } { n ! } x ( x - 1 ) ( x - 2 ) \cdots ( x - n + 1 )$. So we have
$$p _ { 0 } ( x ) = 1 \quad , \quad p _ { 1 } ( x ) = x \quad , \quad p _ { 2 } ( x ) = \frac { x ( x - 1 ) } { 2 } \quad , \quad p _ { 3 } ( x ) = \frac { x ( x - 1 ) ( x - 2 ) } { 3 ! } \quad \ldots$$
Show that for any polynomial $f$ of degree $n$, there exist unique real numbers $b _ { 0 } , b _ { 1 } , \ldots , b _ { n }$ such that $f ( x ) = \sum _ { i = 0 } ^ { n } b _ { i } p _ { i } ( x )$.
(iii) Now suppose that $f ( x )$ is a polynomial such that for each integer $m , f ( m )$ is also an integer. Using the above parts (or otherwise), show that for such $f$, the $b _ { i }$ obtained in part (ii) are integers.