Stirling Number of the Second Kind of 0

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

• 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

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(48)$