Consider the weight function \[ p(x)= \begin{cases} {1\over {\pi}} \sqrt{{\prod_{j=1}^g (x-\alpha_j)} \over{(1-x^2)\prod_{j=1}^g (x-\beta_j)}}&\text{ for } x\in E \\ 0 &\text{ otherwise}\end{cases} \] where \(E\) is the union of \(g+1\) disjoint intervals, \( E=[-1, \alpha_1] \bigcup_{j=1}^{g-1} [\beta_j, \alpha_{j+1}]\bigcup [\beta_g, 1]\), \(-1 < \alpha_j <\beta_j < +1\), \(j=1,\dots, g\). Link to \(E\) the compact Riemann surface \(\mathcal R\) that is given by the hyperelliptic curve \[ y^2 =(x^2-1)\prod_{j=1}^g (x-\alpha_j)(x-\beta_j). \] Let \(\{ P_n\}\) be the sequence of the monic polynomials which are orthogonal with respect to the weight \(p\).
The authors study the system \(\{ P_n\}\) by using the results about functions and differentials on the surface \(\mathcal R\). In particular they construct the representation of \(P_n\) in terms of the Riemann theta function of \(\mathcal R\) and using the Riemann-Roch theorem find the second-order ordinary differential equation satisfied by \(P_n\) and \(Q_n / w\) (where \(Q_n\) are polynomials of the second kind with respect to the weight \(p\) and \(w\) is a Stieltjes transform of \(p\)).
The results are formulated in the explicit form and the author’s methods are deep and interesting (in the reviewer’s opinion). The related approach was used for the first time in approximation theory by N. I. Akhiezer [N. I. Achyeser, “Über einige Funktionen, die in gegebenen Intervallen am wenigsten von Null abweichen”, Bull. Soc. Phys.-Matem. Kazan. Ser. 3, 1–69 (1928; JFM 57.1430.02)]. See also A. L. Lukashov [J. Approx. Theory 95, No. 3, 333–352 (1998; Zbl 0926.41012)].