Stirling Number of n with n-m is Polynomial in n of Degree 2m

Theorem

Unsigned Stirling Number of the First Kind

Let $m \in \Z_{\ge 0}$.

The unsigned Stirling number of the first kind $\ds {n \brack n - m}$ is a polynomial in $n$ of degree $2 m$.

Stirling Number of the Second Kind

Let $m \in \Z_{\ge 0}$.

The Stirling number of the second kind $\ds {n \brace n - m}$ is a polynomial in $n$ of degree $2 m$.

Sources

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