Signed Stirling Number of the First Kind of Number with Greater

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:

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

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 $\map s {n, k}$ of $x^k$ where $k > n$ is $0$.

$\blacksquare$

Also see

• Unsigned Stirling Number of the First Kind of Number with Greater
• Stirling Number of the Second Kind of Number with Greater