I'm looking for reference to literature on the subject of orthogonal polynomials. I am specifically intersted with the following problem:
Let $\omega$ be a compactly supported density function on $\mathbb{R}$, contionuous except perhaps on the edges of it's support. I define an "inner product":
$$ \langle f,g \rangle_{\omega}:= \int_{\mathbb{R}} f(t)g(t)\omega(t)dt$$
I want to find a sequence of polynomials, $\{ P_n \}_{n=0}^\infty $, such that:
(i) Orthogonal with respect to this inner product, $\langle P_m,P_n\rangle_{\omega}=0$ if $m\neq n$.
(ii) $\deg(P_n)=n$ for all $n\in \mathbb{N}_0$.
(iii) For $\text{supp}(\omega)\subseteq[a,b]$, $\{ P_n \}$ is dense in $L^2[a,b]$.
I do not have too much previous knowledge on the subject, so I hope there are literature which assume an introduction to the subject.
So classical references on orthogonal polynomials are: "Classical and Quantum orthogonal polynomials in one variable" by M E H Ismail and "Orthogonal Polynomials" by G Szegö.
From your comment it seems you are interested in explicit expressions so "Hypergeometric Orthogonal Polynomials and Their q-Analogues" by Roelof Koekoek, Peter A. Lesky, René F. Swarttouw could be especially useful.
If this is all a bit too much information I can recommend the introductory part of "Orthogonal Polynomials and Painlevé Equations" by Walter Van Assche, to get a quick idea.