Derivation for Rodrigues formula (orthogonal polynomials)

[J Hill]

Okay, so given a family of orthogonal polynomials under a weight w(x) is described by the differential equation

$Q(x) f'' + L(x) f' + \lambda f$ = 0, where Q(x) is a quadratic (at most) and L(x) is linear (at most).

with the inner product

$\langle f | g \rangle \equiv \int_X f^*(x) g(x) w(x) dx$, it is known that

$f_n(x) = \frac{a_n}{w(x)} \frac{d^n}{dx^n} \Big ( Q^n(x) w(x) \Big)$

Now, I was hoping that someone might be familiar with the derivation of this general form of the Rodrigues formula-- or is it the case that it was just generalized from more specific examples (such as the Legendre polynomials, etc.)?

 

[Office_Shredder]

http://en.wikipedia.org/wiki/Rodrigues'_formula

from the wikipedia article it sounds like it was generalized from the Legendre polynomial solution