Signed Stirling Number of the First Kind of Number with Self

Theorem

$\map s {n, n} = 1$

where $\map s {n, n}$ denotes a signed Stirling number of the first kind.

Proof

From Relation between Signed and Unsigned Stirling Numbers of the First Kind:

$\ds {n \brack n} = \paren {-1}^{n + n} \map s {n, n}$

We have that:

$\paren {-1}^{n + n} = \paren {-1}^{2 n} = 1$

and so:

$\ds {n \brack n} = \map s {n, n}$

The result follows from Unsigned Stirling Number of the First Kind of Number with Self.

$\blacksquare$

Also see

• Unsigned Stirling Number of the First Kind of Number with Self
• Stirling Number of the Second Kind of Number with Self

• Particular Values of Signed Stirling Numbers of the First Kind