On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials

Details On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials

2009-03-15

arxiv.org/abs/0903.2614v1

Mathematics

Classical Analysis and ODEs

24 pages, 6 figures

Scientific paper

We investigate the strong asymptotics of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. The solution is given in terms of critical measures (saddle points of the weighted logarithmic energy on the plane), that are tightly related to quadratic differentials with closed trajectories on the plane. The paper is a continuation of the research initiated in [arXiv:0902.0193]. However, the starting point here is the WKB method, which allows to obtain the strong asymptotics.

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