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

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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

$\blacksquare$


Sources