A few generating functions are tabulated below. Plese see the WU on Bessel functions for an example of how generating functions may be used to immense benefit.
Bessel functions:sum(Jntn) = e(x/2)*(t-1/t)
Hermite polynomials:sum(Hntn/n!)=e-t^2 + 2tx
Note the extra n! in the relation for Hermite polynomials. It is put there by convention. Arfken or Abramowitz would provide a good introduction to Special Functions in general.