Stirling Number of the Second Kind of 0
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Theorem
- $\ds {0 \brace n} = \delta_{0 n}$
where:
- $\ds {0 \brace n}$ 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:
$\ds x^{\underline 0} = \sum_k {0 \brace k} x^k$
Thus we have:
\(\ds x^0\) | \(=\) | \(\ds 1\) | Definition of Integer Power | |||||||||||
\(\ds \) | \(=\) | \(\ds x^{\underline 0}\) | Number to Power of Zero Falling is One |
Thus, in the expression:
- $\ds x^0 = \sum_k {0 \brace k} x^{\underline k}$
we have:
- $\ds {0 \brace 0} = 1$
and for all $k \in \Z_{>0}$:
- $\ds {0 \brace k} = 0$
That is:
- $\ds {0 \brace k} = \delta_{0 k}$
$\blacksquare$
Also see
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(48)$