FoCM 2014 conference

Workshop C5 - Special Functions and Orthogonal Polynomials

December 18, 16:00 ~ 16:30 - Room A21

## Kissing polynomials

### University of Cambridge, United Kingdom   -   ai@damtp.cam.ac.uk

Our point of departure are Gauss-like methods for highly oscillatory quadrature of integrals $\int_{-1}^1 f(x) e^{\mathrm{i}\omega x} \mathrm{d}x$, $\omega\gg1$, employing orthogonal polynomials with a complex-valued weight function. Such polynomials pose fascinating questions of existence and their zeros, plotted in the complex plane as a function of the underlying frequency $\omega$, exhibit an intriguing pattern of "kissing". All this will be analysed and elucidated in the talk. Our main tool is asymptotic analysis for $\omega\gg1$ of Hankel determinants, represented by multivariate highly oscillatory integrals, and their generalisations.

Joint work with Andreas Asheim (NTNU Trondheim), Alfredo Deaño (KU Leuven) and Daan Huybrechs (KU Leuven).