A source tells me that a set $P_n: \mathbb{C} \to \mathbb{C}$ of polynomials are orthogonal over a contour if $$ \frac{1}{2\pi} \int_{\Gamma} P_n(z) \overline{P_m(z)}h(z)d|z|=\delta_{nm}, $$ where h is the weight...
Could someone please continue this sentence and introduce the concept of orthogonal polynomials over contours. Possibly provide an example or refer me to some suitable source. Do examples usually come with one contour or several for the same set of polynomials?
An important example is $\Gamma=S^1$ (the unit circle) and $h(z)=1$. Then $P_n(z)=z^n$ satisfy your requirement. Since the integrand is not holomorphic (there is a complex conjugation involved) and you integrate against $d|z|$ measure, this will no longer be true if you deform $\Gamma$.
Another important example is given by $\Gamma=\mathbb R$, $h=e^{-x^2}$ and Hermite polynomials.
All these are useful for example because they can be used to solve important differential equations, e.g. Laplace equation in various domains.