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

Theorem

$\ds \sum_k {n \brack k} x^k = x^{\overline n}$

where:

$x^{\overline n}$ denotes $x$ to the $n$ rising.

$\ds \sum_k \paren {-1}^{n - k} {n \brack k} x^k$

$=$

$\ds x^{\underline n}$

$\ds \leadsto \ \ $

$\ds \sum_k \paren {-1}^{n - k} {n \brack k} \paren {-x}^k$

$=$

$\ds \paren {-x}^{\underline n}$

putting $-x$ for $x$

$\ds \leadsto \ \ $

$\ds \paren {-1}^n \sum_k {n \brack k} x^k$

$=$

$\ds \paren {-x}^{\underline n}$

$\ds \leadsto \ \ $

$\ds \sum_k {n \brack k} x^k$

$=$

$\ds \paren {-1}^n \paren {-x}^{\underline n}$

$\ds $

$=$

$\ds x^{\overline n}$

$\blacksquare$