The Stirling Numbers of the Second Kind can be found through the ordinary generating function
$$\frac{x^k}{(1-x)(1-2x)...(1-kx)}=\sum_{n=0}^\infty {S(n,k)x^n}$$
Clearly here the g.f is for some fixed positive integer $k$. It has been shown by Gessel (and others probably) that
$$\frac{1}{(1-x)(1-2x)...(1-mx)}=\sum_{n=0}^\infty {S(m+n,n)x^n}$$
It appears here that the fixed integer is no longer fixed, so how can this be valid? My initial thoughts were this: Dividing by $x^k$ in the first equation yields
$$\frac{1}{(1-x)(1-2x)...(1-kx)}=\sum_{n=0}^\infty {S(n,k)x^{n-k}}$$
and by a change of variables, this will equal
$$\frac{1}{(1-x)(1-2x)...(1-kx)}=\sum_{n=0}^\infty {S(n+k,k)x^n}$$
which is close, but we are still seeing a fixed $k$ as the second input. Does anyone know why this is valid for an indexed input?