Zeros of integer function

Consider $w(i) = \sum_{j=0}^{i-1} (-1)^j(i-j)^n{i \choose j}$
where $i=1,2,3,...$ and $n=1,2,3,...$
Question: Prove $w(i)=0$ if $i>n$ .
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Furthermore, suppose ${f}_{i}(x) = \sum_{j=0}^{i-1} (-1)^j(i-j)^x{i \choose j}$
where $i=1,2,3,...$ and $x \in \mathbb{R}$
Question: Are $x=1,2,3,...,i-1$ the only zeros of ${f}_{i}(x)$ ?

Motivation for $w(i)$ see here: http://www.mathhelpforum.com/math-he...ty-194323.html where it's defined recursively.