Signed Stirling Number of the First Kind of Number with Greater

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Theorem

Let $n, k \in \Z_{\ge 0}$

Let $k > n$.


Let $\map s {n, k}$ denote a signed Stirling number of the first kind.

Then:

$\map s {n, k} = 0$


Proof

By definition, the signed Stirling numbers of the first kind are defined as the polynomial coefficients $\map s {n, k}$ which satisfy the equation:

$\displaystyle x^{\underline n} = \sum_k \map s {n, k} x^k$

where $x^{\underline n}$ denotes the $n$th falling factorial of $x$.


Both of the expressions on the left hand side and right hand side are polynomials in $x$ of degree $n$.

Hence the coefficient $s \left({n, k}\right)$ of $x^k$ where $k > n$ is $0$.

$\blacksquare$


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