This survey paper focuses its attention on the following types of orthogonality:
(1) with respect to a non-negative measure \(d\lambda (t)\) on the real line (Hermitean inner product),
(2) with respect to a finite non-negative measure \(d\mu (\theta)\) on the unit circle (Hermitean inner product),
(3) with respect to a complex weight function on the semi-circle (non-Hermitean inner product: \(\int^\pi_0 f(e^{i\theta}) g(e^{i\theta}) w(e^{i\theta}) d\theta)\),
(4) with respect to a complex weight function on a circular arc \(\gamma= \{z:z= -iR+ e^{i\theta} \sqrt {(R^2+1)}\), \(\varphi\leq \theta \leq\pi- \varphi\), \(\tan \varphi =R\}\) (non-Hermitean inner product: \(\int_\gamma f(z)g(z) w(z) (iz-R)^{-1}dz)\) and the situation where the arc is replaced by its mirror image in the real axis, \(iz-R\) is replaced by \(iz+R\),
(5) Geronimus’ version (starting from \(\{p_k\}\) of type 1 on a finite interval, a weight function \(\chi(z)\) having one or more singularities inside a simple curve \(C\) is found, such that \(\int_C p_k(z) p_m(z) \chi(z)dz\) \(=h_m\delta_{km})\) extended to the situation where the starting sequence is of type 3 or 4,
(6) with respect to a weight function on a number of rays in the complex plane, in the paper restricted to \(2m\) rays \(\ell(s)\) from the origin to the \(2m\)th roots of unity \(\varepsilon_s = e^{i \pi s/m} (0\leq s\leq 2m-1)\) (inner product \((f,g)= \sum^{2m-1}_{s=0} \varepsilon_s^{-1} \int_{\ell(s)} f(z) \overline {g(z)} |w(z)|dz)\).
Types 1,2 together take 4 pages, type 3 uses 5 pages, type 4 a little more than 3 pages, type 5 about 2 1/2 pages and type 6 around 8 pages.
The information centers around (three term) recurrence relations, norms and location of zeros. Moreover, in case 6, the connection with incomplete lacunary polynomials is established.
The list of references contains 43 items (the classification paper by Hahn is missing) and it becomes quite clear that the prolific field of orthogonal polynomials can not be treated in a survey paper anymore: the OP community indeed needs a type of “Bateman project’ (just think about \(q\)-orthogonality, Sobolev inner products in discrete and continuous setting both, etc.).