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Derivation for Rodrigues formula (orthogonal polynomials)

  1. Apr 27, 2012 #1
    Okay, so given a family of orthogonal polynomials under a weight w(x) is described by the differential equation

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

    with the inner product

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

    [itex]f_n(x) = \frac{a_n}{w(x)} \frac{d^n}{dx^n} \Big ( Q^n(x) w(x) \Big) [/itex]

    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.)?
  2. jcsd
  3. Apr 28, 2012 #2


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