Let $\lbrace P\_n(z)\rbrace\_{n\in\mathbb N\_0}$ be a family of polynomials defined by a generating function $g(t,z)=\sum\limits\_{n=0}^\infty P\_n(z)t^n$ or by a contour integral $P\_n(z)=\frac1{2\pi i}\oint\frac{g(t,z)}{t^{n+1}}dt$.
Are there known sufficient conditions on $g$ or on the $P\_n$ themselves that guarantee the existence of a weight function $w:I\to \mathbb R^+\_0$ (where $I\subset\mathbb R$ is an appropriate interval) such that the $P\_n$ are orthogonal w.r.t. $w$?



When can a family of polynomials get a weight function to be made orthogonal?
Favard's theorem characterizes this in terms of the three-term recurrence. Suppose the polynomials $P\_n$ are normalized so that they are monic. Then they are orthogonal polynomials with respect to some Borel measure if and only if there are constants $\alpha\_n$ and $\beta\_n$ such that $P\_n(x) = (x+\alpha\_n) P\_{n-1}(x) + \beta\_n P\_{n-2}(x)$ and $\beta\_n < 0$. (The sign condition on $\beta\_n$ is needed to get a positive measure. I think you still get a signed measure if you have a three-term recurrence with $\beta\_n \ge 0$, but I'm not certain offhand.)


This is pretty easy to test for in practice if you are given a sequence of polynomials numerically. Strictly speaking, it doesn't guarantee a weight function as specified in your question, since the measure may not be absolutely continuous with respect to Lebesgue measure, but I assume that's not what you really care about. If it is, then I'm not sure offhand how to characterize that case.


