On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
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DOI：
10.4153/cjm-2006-030-x
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发表时间：
2006-08
期刊：
影响因子：
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通讯作者：
Y. Chiang;M. Ismail
中科院分区：
文献类型：
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作者：
Y. Chiang;M. Ismail
本文证明了Bank，Laine和兰利研究的一类二阶周期常微分方程解的值分布（复振荡）与合流超几何函数，Bessel函数和Bessel多项式密切相关.作为结果，我们给出了两类常微分方程解的Nevanlinna理论意义下的零分布的完整刻画。我们的方法使用特殊的功能和他们的渐近。作为结果，还得到了关于贝塞尔和库仑波函数关于参数的零点（有限零点）问题的有限性的新结果。我们证明了上述“特殊函数方法”未涵盖的其余常微分方程类的问题可以用一个经典的允许多项式解的微分方程海涅问题来描述。
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.