Remembering Special Function Equations? I have an awful memory when it comes to factoids, I need to remember the Legendre, Hermite, Laguerre, Chebyshev, Hypergeometric & Jacobi equations, all of which are of the form $p(x)y'' + q(x)y' + r(x)y = 0$, where $p$ is a second degree polynomial, $q$ is a first degree polynomial & $r$ is a zero'th degree polynomial (interpreted as an eigenvalue).
Now, I can derive $r(x)$ by following Arfken's development & just substituting in a series solution & deriving what the eigenvalue should be.
Thus I'm left with finding out a way to remember the coefficients $p(x)$ & $q(x)$ for the Legendre, Hermite, Laguerre, Chebyshev, Hypergeometric & Jacobi equations. Is there any simple way to do this? Any unifying procedure or thought process?
Considering that this question could have been asked by including the $r(x)$ term, would you have recommended subbing in the series to derive the eigenvalue, or just told me to remember them all? :tongue:
I won't start asking about ways to remember things like Bessel's equation, Weber, Matthieu, Lame etc... as I feel that is a fruitless task, though any nice ideas or tips on these monstrosities would be greatly appreciated.
 A: Unfortunately, I am afraid that there is no other way than learning it by heart.
The relationships between the name of the Mathematicians and the special functions and related equations are historical. There is no logical naming system and classification which could help for memorizing the background.
This question has been considered in the paper "Safari in the Country of Special Functions" (Comment at end of section 5, p.28). Published on Scribd :
http://www.scribd.com/JJacquelin/documents 
A: The special functions you mention arose as the solutions to certain ODEs/eigenvalue problems which in turn arose (in many cases) from solving classical boundary value problems for PDE (heat, wave, Laplace's equation) in various coordinate systems by the method of separation of variables. Because of these, the form that the equations take is dictated by the physics of the particular motivating problem. (Note that this is not the only way motivate these equations, but it is a very good way, IMO.)
As far as unification, you could think of them as particular Sturm-Liouville problems where appropriate.
