Let $c(n,k) = |s(n,k)| = (-1)^{n-k} s(n,k)$. It is well known that for fixed $k$ $S(n+k,n)$ and $s(n+k,n)$ are polynomials in $n$ of degree $2k$ with leading coefficient $(2k-1)!!/(2k)!$. If we extend these polynomials to all values of $n$ then $S(-n+k,-n)=c(n,n-k)$, or with appropriate restrictions on $n$ and $k$, $s(k,n)= c(-n,-k)$. Similar results apply to inverse pairs that come from exponential generating functions for powers of compositional inverse generating functions (for Stirling numbers these are $e^x-1$ and $\log(1+x)$). This is a consequence of Lagrange inversion, or more precisely, a form of Lagrange inversion called the Schur-Jabotinsky theorem.

For more on the duality between Stirling numbers of the first and second kinds, see I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory Ser. A 24 (1978), 24–33, and D. E. Knuth, Two notes on notation, American Mathematical Monthly 99 (1992),403–422.

Let $c(n,k) = |s(n,k)| = (-1)^{n-k} s(n,k)$. It is well known that for fixed $k$, $S(n+k,n)$ and $s(n+k,n)$ are polynomials in $n$ of degree $2k$ with leading coefficient $(2k-1)!!/(2k)!$. If we extend these polynomials to all values of $n$ then $S(-n+k,-n)=c(n,n-k)$, or with appropriate restrictions on $n$ and $k$, $s(k,n)= c(-n,-k)$. Similar results apply to inverse pairs that come from exponential generating functions for powers of compositional inverse generating functions (for Stirling numbers these are $e^x-1$ and $\log(1+x)$). This is a consequence of Lagrange inversion, or more precisely, a form of Lagrange inversion called the Schur-Jabotinsky theorem.

For more on the duality between Stirling numbers of the first and second kinds, see I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory Ser. A 24 (1978), 24–33, and D. E. Knuth, Two notes on notation, American Mathematical Monthly 99 (1992), 403–422.