Stirling Number of the Second Kind of 0

Theorem

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

where:
 * $\displaystyle \left\{ {0 \atop n}\right\}$ denotes a Stirling number of the second kind
 * $\delta_{0 n}$ denotes the Kronecker delta.

Proof
By definition of Stirling numbers of the second kind:

$\displaystyle x^{\underline 0} = \sum_k \left\{ {0 \atop k}\right\} x^k$

Thus we have:

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

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:
 * $\displaystyle \left\{{0 \atop k}\right\} = \delta_{0 k}$

Also see

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


 * Particular Values of Stirling Numbers of the Second Kind