  For use in calculating higher-order coherent- and squeezed- state quantities,
we derive generalized generating functions for the Hermite polynomials. They
are given by $\sum_{n=0}^{\infty}z^{jn+k}H_{jn+k}(x)/(jn+k)!$, for arbitrary
integers $j\geq 1$ and $k\geq 0$. Along the way, the sums with the Hermite
polynomials replaced by unity are also obtained. We also evaluate the action of
the operators $\exp[a^j(d/dx)^j]$ on well-behaved functions and apply them to
obtain other sums.
