I am asking a simple conceptual question. I saw in many Mathematics and Mathematical physics text books that the Legendre polynomials and Laguerre polynomials "falling from the sky"! I mean, I didn't get the concept behind and how those polynomials derived. The textbooks rather says a second order differential equation and says the solution is in this form. I wish to know why people assuming this kind of differential equations and also what is the physical concept behind. It would be rather interesting to know how basically the solutions for those differential equations derived.
Or please tell me a text book/reference where I can find those concepts
Thanks a lot.
The basic concept: when we work with a Hilbert space of functions, we want to have an orthonormal basis that consists of particularly simple functions. The trigonometric (or complex exponential) basis works very well for $L^2([a,b])$. But algebraic polynomials can serve this purpose too. Since polynomials are not integrable over unbounded intervals, one includes a weight such as $w(x)=e^{-x}$ on $[0,\infty)$ or $w(x)=e^{-x^2}$ on $\mathbb R$. Then the monomials $1,x,x^2,\dots$ are elements of the weighted Lebesgue space $L^2_w = \{f:\int |f|^2 w<\infty\}$. Applying the Gram-Schmidt process to the sequence of monomials produces an orthonormal sequence, which in many cases turns out to be complete, i.e., an orthonormal basis. With the weights I mentioned above we get the Laguerre and Hermite polynomials, respectively.
Orthogonal polynomials have a large number of interesting properties, many of which are easily proved by induction. Recurrence relations, for example: polynomials of different degrees turn out to be related via linear combinations involving derivatives. Also, they usually satisfy a simple differential equation with degree $n$ as parameter, which can be used as a shortcut to quickly define the polynomials (without going through Gram-Schmidt, etc).
Further reading: you can start with the short Wikipedia article Orthogonal polynomials and proceed to the references listed there, out of which I recommend the classical book by Szegő, Orthogonal polynomials.