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ON THE EXISTENCE OF FINITE CRITICAL TRAJECTORIES IN A FAMILY OF QUADRATIC DIFFERENTIALS

Part of: Geometric function theory Asymptotic theory Two-dimensional theory

Published online by Cambridge University Press:  16 March 2016

FAOUZI THABET
FAOUZI THABET*
Affiliation:
Institut Supérieur des Sciences Appliquées et de Technologie de Gabès, Avenue Omar Ibn El Khattab, 6029, Tunisia email faouzithabet@yahoo.fr
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Abstract

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We discuss the existence of finite critical trajectories connecting two zeros in certain families of quadratic differentials. In addition, we reprove some results about the support of the limiting root-counting measures of the generalised Laguerre and Jacobi polynomials with varying parameters.

Keywords

quadratic differentialstrajectories and orthogonal trajectorieshomotopic classesgeodesicsLaguerre and Jacobi polynomials

MSC classification

Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
Secondary: 31A35: Connections with differential equations 34E05: Asymptotic expansions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 80 - 91
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Atia, M. J., Martinez-Finkelshtein, A., Martinez-Gonzalez, P. and Thabet, F., ‘Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters’, J. Math. Anal. Appl. 416 (2014), 52–80.Google Scholar
Bagrov, V. G. and Gitman, D. M., Exact Solutions of Relativistic Wave Equations (Kluwer Academic Publications, Dordrecht, 1990).Google Scholar
Gonchar, A. A. and Rakhmanov, E. A., ‘Equilibrium measure and the distribution of zeros of extremal polynomials’, Mat. Sb. 53 (1986), 119–130.Google Scholar
Kuijlaars, A. B. J. and Martinez-Finkelshtein, A., ‘Strong asymptotics for Jacobi polynomials with varying nonstandard parameters’, J. Anal. Math. 94 (2004), 195–234.Google Scholar
Kuijlaars, A. B. J., Martinez-Finkelshtein, A. and Orive, R., ‘Orthogonality of Jacobi polynomials with general parameters’, Electron. Trans. Numer. Anal. 19 (2005), 1–17.Google Scholar
Kuijlaars, A. B. J. and McLaughlin, K. T.-R., ‘Asymptotic zero behavior of Laguerre polynomials with negative parameter’, Constr. Approx. 20(4) (2004), 497–523.Google Scholar
Martinez-Finkelshtein, A., Martinez-Gonzalez, P. and Orive, R., ‘On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters’, J. Comput. Appl. Math. 133 (2001), 477–487.Google Scholar
Martinez-Finkelshtein, A., Martinez-Gonzalez, P. and Thabet, F., ‘Trajectories of quadratic differentials for Jacobi polynomials with complex parameters’, Comput. Methods Funct. Theory, to appear. Published online (8 December 2015). doi:10.1007/s40315-015-0146-7.Google Scholar
Shapiro, B., ‘On Evgrafov-Fedoryuk’s theory and quadratic differentials’, Anal. Math. Phys. 5 (2015), 171–181.Google Scholar
Stahl, H., ‘Orthogonal polynomials with complex-valued weight function. I, II’, Constr. Approx. 2(3) (1986), 225–251.Google Scholar
Strebel, K., Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 5 (Springer, Berlin, 1984).Google Scholar
Szego, G., Orthogonal Polynomials, 4th edn, American Mathematical Society Colloquium Publications, 23 (American Mathematical Society, Providence, RI, 1975).Google Scholar