Unsigned Stirling Number of the First Kind of 1

Theorem

$\ds {1 \brack n} = \delta_{1 n}$

where:

$\ds {1 \brack n}$ denotes an unsigned Stirling number of the first kind

$\delta_{1 n}$ is the Kronecker delta.

Proof

\(\ds {1 \brack n}\)

\(=\)

\(\ds 0 \times {0 \brack n} + {0 \brack n - 1}\)

Definition of Unsigned Stirling Numbers of the First Kind

\(\ds \)

\(=\)

\(\ds {0 \brack n - 1}\)

\(\ds \)

\(=\)

\(\ds \delta_{0 \paren {n - 1} }\)

Definition of Unsigned Stirling Numbers of the First Kind

\(\ds \)

\(=\)

\(\ds \delta_{1 n}\)

$0 = n - 1 \iff 1 = n$

$\blacksquare$

Also see

• Signed Stirling Number of the First Kind of 1
• Stirling Number of the Second Kind of 1

• Particular Values of Unsigned Stirling Numbers of the First Kind