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(J_{n}t^{n}) = e^{(x/2)*(t-1/t)}

Legendre Polynomials:sum(P_{n}t^{n})=(1-2tx+t^{2})^{(-1/2)}

Laguerre Polynomials:sum(L_{n}t^{n})=
(1-t)^{(-1)}e^{-(xt/1-t)}

Hermite polynomials:sum(H_{n}t^{n}/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.