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

  Published:2006-08-01
 Printed: Aug 2006
  • Yik-Man Chiang
  • Mourad E. H. Ismail
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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 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: Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Proble Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Proble
MSC Classifications: 34M10, 33C15, 33C47 show english descriptions Oscillation, growth of solutions
Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
Other special orthogonal polynomials and functions
34M10 - Oscillation, growth of solutions
33C15 - Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
33C47 - Other special orthogonal polynomials and functions
 

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