On value distribution theory of second order periodic ODEs, special functions and orthogonal polynomials
Abbreviated Journal Title
Can. J. Math.-J. Can. Math.
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
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.
Canadian Journal of Mathematics-Journal Canadien De Mathematiques
"On value distribution theory of second order periodic ODEs, special functions and orthogonal polynomials" (2006). Faculty Bibliography 2000s. 4673.