This is a beautiful paper connecting deep mathematical results with problems in mathematical physics.
The author restricts himself to semi-classical orthogonal polynomials (i.e., polynomials orthogonal – may be on a set of arcs – with respect to a weight function for which is a rational function). The coefficients of the well-known three term recurrence relation satisfy nonlinear differential equations with respect to a certain parameter (here the work on monodromy by the Chudnovskij brothers plays a role (references – in the paper).
The author treats five examples in varying detail:
1. Jacobi weight with three factors .
2. on .
3. on (simplest non-trivial Freud weight).
4. on (: Maxwell polynomials).
The paper contains an extensive list of references (93 items).