We consider the formal and analytic aspects of the strong asymptotics of pseudo-orthogonal polynomials (i.e. “orthogonal” with respect to a complex measure). On a formal level the asymptotics is intimately related to the classification of commuting pairs of difference operators one of which is symmetric and tridiagonal. This in turn realizes the common solutions of these difference operators in terms of sections of line bundles on a Riemann surface (hyperelliptic). From geometrical considerations stemming from the Riemann-Hilbert analysis we show that the line bundle which is relevant to the asymptotics is a spinor bundle.

In a second stage we show that the heuristics is sound and gives the actual strong asymptotics using the nonlinear steepest-descent method developed by Fokas-Its-Deift-Zhou; the main ingredient is the ability of constructing hyperelliptic Riemann surfaces with a prescribed harmonic real function of definite signs. Such construction impinges on the notion of quadratic differentials and their theory, developed by K. Strebel and used in a different context for associating Riemann surfaces to decorated ribbon graphs.
(Based on a work with M.Y. Mo.)

Dept. of Math. & Stat., Concordia University, Montreal