Signed Stirling Number of the First Kind of 0

Theorem

 * $s \left({0, n}\right) = \delta_{0 n}$

where:
 * $s \left({0, n}\right)$ denotes a signed Stirling number of the first kind
 * $\delta_{0 n}$ denotes the Kronecker delta.

Proof
By definition of signed Stirling number of the first kind:

$\displaystyle x^{\underline 0} = \sum_k s \left({0, k}\right) x^k$

Thus we have:

Thus, in the expression:
 * $\displaystyle x^{\underline 0} = \sum_k s \left({0, k}\right) x^k$

we have:
 * $s \left({0, 0}\right) = 1$

and for all $k \in \Z_{>0}$:
 * $s \left({0, k}\right) = 0$

That is:
 * $s \left({0, k}\right) = \delta_{0 k}$

Also see

 * Unsigned Stirling Number of the First Kind of 0
 * Stirling Number of the Second Kind of 0


 * Particular Values of Signed Stirling Numbers of the First Kind