generating function

A whole infinite family of polynomials — P_0, P_1, P_2, and so on — is a lot to carry around. A generating function packs the entire family into one tidy function of two variables: expand it as a power series in an auxiliary variable t, and the coefficient of t^n is the n-th member of the family. It is a single master function from which you can shake out every polynomial on demand.

Concretely, a function G(x, t) is a generating function for a polynomial family p_n(x) if its Taylor expansion in t reads G(x, t) = sum over n>=0 of p_n(x) t^n (sometimes with an n! in the denominator). For the Legendre polynomials the generating function is 1/sqrt(1 - 2 x t + t^2) = sum P_n(x) t^n; for the Hermite polynomials it is e^{2 x t - t^2} = sum H_n(x) t^n/n!. To recover, say, P_2(x) you just expand the closed form to order t^2 and read off the coefficient. The remarkable thing is that proofs of the polynomials' recurrence relations, their orthogonality, and special values all follow by differentiating or manipulating this one compact expression, rather than juggling infinitely many separate identities.

These generating functions are not curiosities — they encode real physics. The Legendre generating function 1/sqrt(1 - 2 x t + t^2) is literally the expansion of the Newtonian/Coulomb potential 1/|r - r'| in terms of the ratio of distances, which is why the multipole expansion drops straight out of it. The Hermite and Laguerre generating functions similarly compress the structure of the quantum oscillator and the hydrogen atom. A generating function is, in short, the most economical single object that remembers an entire orthogonal family at once.