In our math lecture we started to deal with "orthogonal polynomials". I have to say that I don't understand some things. Therefore, I did a lot of research on orthogonal polynomials and wrote down the most important things I found.
Definition: Weighted scalar product $\text{A weighted scalar product over the space of the twice integrable function} \; L ^ {2} ([a, b]\; \text{on the interval}\; [a, b] \subseteq \mathbb{R}\; \text{is defined by}\;$ $ (p, q) : = \int \limits_ {a} ^ {b} \omega(x) p(x) q(x) dx$
Theorem: Gram - Schmidt - Algorithm $\text {Let be}\; H\; \text{a unitary space and}\; S \subseteq H\; \text{a finite dimensional subspace.}$ $\text{For every base}\; B: = \{v_ {1}, \ldots, v_ {n} \} \; \text{of}\; S\; \text{we can construct an Orthogonal System with the Gram-Schmidt Algorithm:}$ $ \varphi_{1}:= v_ {1}\; \text{and for}\; k = 2, \ldots, n: \varphi_{k}: = v_{k} - \sum\limits_{i = 1}^{ k - 1} \frac{(v_ {k}, \varphi_{i})}{(\varphi_{i}, \varphi_{i})} \varphi_{i} $
In an exercise, I had to prove the following theorem:
The orthogonal polynomials $p_{k}$ respective to the weighted scalarproduct satisfies $ p_{k} = \frac {C_ {k}} {\omega (x) } \frac {d^{k}} {dx^{k}} \left [\omega(x) (x - a)^{k} (b - x)^{k} \right] $, $ C_{k} \in \mathbb{R}$
This is called the Rodrigues Formula.
I can prove this theorem. I have shown that the polynomials $ p_{k} $ are all orthogonal to each other. But it's still not clear to me how exactly one comes up with these polynomials. So I'm more interested in a derivation of this formula. Is it possible to derive these polynomials in general or its way too complicated? Because I found nowhere a derivation.
How do I get these polynomials? First I would take the monomial basis $ B: = \{v_{0} = 1, v_{1} = x, v_{2} = x^{2}, \ldots \} $ and orthogonalize it with the Gram - Schmidt - Algorithm. I tried that out this morning by trying to calculate $ p_ {1} $. But the result was very different. Does anyone have an idea? I Would be grateful for any input.