Title

On value distribution theory of second order periodic ODEs, special functions and orthogonal polynomials

Authors

Authors

Y. M. Chiang;M. E. H. Ismail

Comments

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Abbreviated Journal Title

Can. J. Math.-J. Can. Math.

Keywords

complex oscillation theory; exponent of convergence of zeros; zero; distribution of Bessel and confluent hypergeometric functions; Lommel; transform; Bessel polynomials; Heine problem; LINEAR-DIFFERENTIAL EQUATIONS; OSCILLATION-THEORY; ZEROS; COULOMB; ORDER; WAVES; Mathematics

Abstract

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 Besset 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.

Journal Title

Canadian Journal of Mathematics-Journal Canadien De Mathematiques

Volume

58

Issue/Number

4

Publication Date

1-1-2006

Document Type

Article

Language

English

First Page

726

Last Page

767

WOS Identifier

WOS:000239417700003

ISSN

0008-414X

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