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On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials

Published online by Cambridge University Press:  20 November 2018

Yik-Man Chiang  and
Mourad E. H. Ismail
Yik-Man Chiang
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, P.R. China e-mail: machiang@ust.hk
Mourad E. H. Ismail
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816, U.S.A. e-mail: ismail@math.ucf.edu
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Abstract

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We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above “special function approach” can be described by a classical Heine problem for differential equations that admit polynomial solutions.

Keywords

34M1033C1533C47Complex Oscillation theoryExponent of convergence of zeroszero distribution of Bessel and Confluent hypergeometric functionsLommel transformBessel polynomialsHeine Problem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 4 , 01 August 2006 , pp. 726 - 767
Copyright
Copyright © Canadian Mathematical Society 2006

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