# Sum over k of Unsigned Stirling Numbers of First Kind by x^k

## Theorem

$\displaystyle \sum_k \left[{n \atop k}\right] x^k = x^{\overline n}$

where:

$\displaystyle \left[{n \atop k}\right]$ denotes an unsigned Stirling number of the first kind
$x^{\overline n}$ denotes $x$ to the $n$ rising.

## Proof

 $\displaystyle \sum_k \left({-1}\right)^{n - k} \left[{n \atop k}\right] x^k$ $=$ $\displaystyle x^{\underline n}$ Definition of Unsigned Stirling Numbers of the First Kind $\displaystyle \leadsto \ \$ $\displaystyle \sum_k \left({-1}\right)^{n - k} \left[{n \atop k}\right] \left({-x}\right)^k$ $=$ $\displaystyle \left({-x}\right)^{\underline n}$ putting $-x$ for $x$ $\displaystyle \leadsto \ \$ $\displaystyle \left({-1}\right)^n \sum_k \left[{n \atop k}\right] x^k$ $=$ $\displaystyle \left({-x}\right)^{\underline n}$ $\displaystyle \leadsto \ \$ $\displaystyle \sum_k \left[{n \atop k}\right] x^k$ $=$ $\displaystyle \left({-1}\right)^n \left({-x}\right)^{\underline n}$ $\displaystyle$ $=$ $\displaystyle x^{\overline n}$ Rising Factorial in terms of Falling Factorial of Negative

$\blacksquare$