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$S_1(n,0)=S_1(0,n)= \delta_n \; \;$ and, for $n > 0$,

$$S_1(n,k)= \lim_{y \to 0} \; \frac{y^{-k}}{k!} \; \sum_{j=1}^k (-1)^j \binom{k}{j} \; \frac{(-j \; y)!}{(-j \; y-n)!} \;$$

$$ = \lim_{y \to 0} \; \frac{y^{-k}}{k!} \; \sum_{j=1}^k (-1)^{n-j} \binom{k}{j} \; \frac{(j \; y - 1 + n)!}{(j \; y-1)!} \; .$$

For a derivation, see A class of differential operators and the Stirling numbers. Note that with $y$ small enough taking the nearest integer generates $S_1$.

$S_1(n,0)=S_1(0,n)= \delta_n \; \;$ and, for $n > 0$,

$$S_1(n,k)= \lim_{y \to 0} \; \frac{y^{-k}}{k!} \; \sum_{j=1}^k (-1)^j \binom{k}{j} \; \frac{(-j \; y)!}{(-j \; y-n)!} \;$$

$$ = \lim_{y \to 0} \; \frac{y^{-k}}{k!} \; \sum_{j=1}^k (-1)^{n-j} \binom{k}{j} \; \frac{(j \; y - 1 + n)!}{(j \; y-1)!} \; $$

$$= \sum_{j=k}^n \; S_1(n,j)\; (-y)^{j-k}\;S_2(j,k) \; |_{y=0} \; . $$

For a derivation, see A class of differential operators and the Stirling numbers. Note that with $y$ small enough taking the nearest integer generates $S_1$.

$S_1(n,0)=S_1(0,n)= \delta_n \; \;$ and, for $n > 0$,

$$S_1(n,k)= \lim_{y \to 0} \sum_{j=1}^k (-1)^j \binom{k}{j} \; \frac{(-j \; y)!}{(-j \; y-n)!} \; \frac{y^{-k}}{k!}$$

$$ = \lim_{y \to 0} \sum_{j=1}^k (-1)^{n-j} \binom{k}{j} \; \frac{(j \; y - 1 + n)!}{(j \; y-1)!} \; \frac{y^{-k}}{k!} \; .$$

For a derivation, see A class of differential operators and the Stirling numbers. Note that with $y$ small enough taking the nearest integer generates $S_1$.

$S_1(n,0)=S_1(0,n)= \delta_n \; \;$ and, for $n > 0$,

$$S_1(n,k)= \lim_{y \to 0} \; \frac{y^{-k}}{k!} \; \sum_{j=1}^k (-1)^j \binom{k}{j} \; \frac{(-j \; y)!}{(-j \; y-n)!} \;$$

$$ = \lim_{y \to 0} \; \frac{y^{-k}}{k!} \; \sum_{j=1}^k (-1)^{n-j} \binom{k}{j} \; \frac{(j \; y - 1 + n)!}{(j \; y-1)!} \; .$$

For a derivation, see A class of differential operators and the Stirling numbers. Note that with $y$ small enough taking the nearest integer generates $S_1$.

$S_1(n,0)=S_1(0,n)= \delta_n \; \;$ and, for $n > 0$,

$$S_1(n,k)= \lim_{y \to 0} \sum_{j=1}^k (-1)^j \binom{k}{j} \; \frac{(-j \; y)!}{(-j \; y-n)!} \; \frac{y^{-k}}{k!}$$

$$ = \lim_{y \to 0} \sum_{j=1}^k (-1)^{n-j} \binom{k}{j} \; \frac{(j \; y - 1 + n)!}{(j \; y-1)!} \; \frac{y^{-k}}{k!} $$

$S_1(n,0)=S_1(0,n)= \delta_n \; \;$ and, for $n > 0$,

$$S_1(n,k)= \lim_{y \to 0} \sum_{j=1}^k (-1)^j \binom{k}{j} \; \frac{(-j \; y)!}{(-j \; y-n)!} \; \frac{y^{-k}}{k!}$$

$$ = \lim_{y \to 0} \sum_{j=1}^k (-1)^{n-j} \binom{k}{j} \; \frac{(j \; y - 1 + n)!}{(j \; y-1)!} \; \frac{y^{-k}}{k!} \; .$$

For a derivation, see A class of differential operators and the Stirling numbers. Note that with $y$ small enough taking the nearest integer generates $S_1$.