Unsigned Stirling Number of the First Kind of 0

Theorem

 * $\displaystyle \left[{0 \atop n}\right] = \delta_{0 n}$

where:
 * $\displaystyle \left[{0 \atop n}\right]$ denotes an unsigned Stirling number of the first kind
 * $\delta_{0 n}$ denotes the Kronecker delta.

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

$\displaystyle x^{\underline 0} = \sum_k \left({-1}\right)^{0 - k} \left[{0 \atop k}\right] x^k$

Thus we have:

Thus, in the expression:
 * $\displaystyle x^{\underline 0} = \sum_k \left({-1}\right)^{-k} \left[{0 \atop k}\right] x^k$

we have:
 * $\displaystyle \left[{0 \atop 0}\right] = 1$

and for all $k \in \Z_{>0}$:
 * $\displaystyle \left[{0 \atop k}\right] = 0$

That is:
 * $\displaystyle \left[{0 \atop k}\right] = \delta_{0 k}$