Exponential generating series of binomial coefficients: $\sum_{k=0}^\infty{ k \choose j}\frac{x^k}{k!} $

Question

I'm wondering if anyone knows what it is? The exponential generating series I have in mind is $$ f_j(x) = \sum_{k=0}^\infty { k \choose j} \dfrac{x^k}{k!}. $$ Thanks!

Answer 1

You may just write

$$ \sum_{k=0}^\infty { k \choose j} \dfrac{x^k}{k!}=\sum_{k=j}^\infty { k \choose j} \dfrac{x^k}{k!}=\sum_{k=j}^\infty \frac{k!}{j!(k-j)!}\dfrac{x^k}{k!}=\frac{x^j}{j!}\sum_{k=j}^\infty \frac{x^{(k-j)}}{(k-j)!}=\frac{x^j}{j!}e^x. $$