Arbitrary Constants in Orthogonal Polynomials?

Question

The generalized Rodrigues formula (Hassani Mathematical Physics P174) is of the form

$$K_n\frac{1}{w}(\frac{d}{dx})^n(wp^n)$$

The constant $K_n$ is seemingly chosen completely arbitrarily, & I really need to be able to figure out a quick way to derive whether it should be $K_n = \tfrac{(-1)^n}{2^nn!}$ in the case of Jacobi polynomials (reducable to Legendre, Chebyshev or Gegenbauer), $K_n = \tfrac{1}{n!}$ for Laguerre polynomials & $K_n = (-1)^n$ for Hermite polynomials. The best I have so far is actually working out the n'th derivative of $(wp^n)$ in the case of Legendre polynomials, but that method becomes crazy with any of the other polynomials & as Hassani says the choices are arbitrary so they probably don't work. My question is, how do I get derive constants without any memorization, whether by some nice trick or by the method one uses to arbitrarily choose their values - I'd really appreciate it, thanks!

Answer 1

The "normalization" or "standardization" constants $K_n$ have historically been chosen differently for differently problems to accomplish different purposes: sometimes it is to make the resulting polynomials orthonormal, sometimes it is to make the leading coefficient in the polynomial to be 1, and sometimes they are rescaled for something specific to the application from which the special ODE arose.

There is a very nice table here which compiles a wide variety of information concerning the classical orthogonal polynomials, including the "standard" selections for the constants $K_n$ in Rodrigues' formula.