The generalized Rodrigues formula (Hassani, Mathematical Physics, p. 174) is of the form
$$p_n=K_n\frac{1}{w}\left(\frac{d}{dx}\right)^n(wp^n)$$
The constant $K_n$ is seemingly chosen completely arbitrarily, and 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 (which covers by extension the Legendre, Chebyshev and Gegenbauer polynomials), $K_n = \tfrac{1}{n!}$ for Laguerre polynomials, and $K_n = (-1)^n$ 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, and as Hassani says the choices are arbitrary so they probably don't work. My question is, how can I derive the constants without any memorization, whether by some nice trick or by the method one uses to arbitrarily choose their values. I'd really appreciate some help.
Best Answer
There is, in general, no general mnemonic for the $K_n $. Those constants are fixed by convention and such conventions are different between the big families of classical OPs, which reflects the different situations they're used in. No instructor should expect you to memorize them. For problem solving, or as a working research phycisist, the usual is to have a copy of Abramowitz & Stegun, or simply the DLMF, to refer to for the precise form of the normalization factors in such formulae.
Edit in response to comment:
To be clear, this is not doable. The conventions are chosen independently for the different families and respond to different pressures; there is no overarching scheme. What Szegö is doing in your quote is reducing the Rodrigues formula to the form $\text{const}\times P_n^{(\alpha,\beta)}$ and then fixing the normalization using the fact that $$ P_n^{(\alpha,\beta)}(1)=\begin{pmatrix}n+\alpha\\n\end{pmatrix}, $$ which is equally arbitrary.
Have a look at the table with the definitions of the different families in the DLMF (the notation for which is here) and at the table of special values. Some are normalized to a constant norm (e.g. Chebyshev), some are normalized to constant leading coefficient (e.g. Hermite $\rm{He}_n$, Chebyshev), some are normalized to a constant value at $x=0$ or $x=1$ (e.g. Legendre). Others are normalized due to other considerations (like the Hermite $H_n$, which are normalized so $H_n(x)e^{-x^2/2}$ is an eigenfunction of the Fourier transform).
What you want is impossible. If you want the normalization constants at your fingertips, memorize them the hard way, or have the DLMF or the NIST Handbook or Abramowitz and Stegun at your fingertips.