generating function (thing)

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_ntⁿ) = e^{(x/2)*(t-1/t)}
Legendre Polynomials:sum(P_ntⁿ)=(1-2tx+t²)^(-1/2)
Laguerre Polynomials:sum(L_ntⁿ)= (1-t)^(-1)e^-(xt/1-t)
Hermite polynomials:sum(H_ntⁿ/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.