- #1
J Hill
- 9
- 0
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.)?
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.)?