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Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture

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Abstract

In this paper, we introduce certain n-th order nonlinear Loewy factorizable algebraic ordinary differential equations for the first time and study the growth of their meromorphic solutions in terms of the Nevanlinna characteristic function. It is shown that for generic cases all their meromorphic solutions are elliptic functions or their degenerations and hence their order of growth is at most two. Moreover, for the second order factorizable algebraic ODEs, all their meromorphic solutions (except for one case) are found explicitly. This allows us to show that a conjecture proposed by Hayman in 1996 holds for these second order ODEs.

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Authors and Affiliations

  1. Department of Mathematics, The University of Hong Kong, Pokfulam Road, Pokfulam, Hong Kong

    Tuen-Wai Ng

  2. Institute for Advanced Study, Shenzhen University, Shenzhen, PR China

    Cheng-Fa Wu

Authors
  1. Tuen-Wai Ng
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  2. Cheng-Fa Wu
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Correspondence to Tuen-Wai Ng.

Additional information

Supported by PROCORE - France/Hong Kong joint research grant, F-HK39/11T and RGC grant HKU 704409P.

Supported by NSFC grant (No. 11701382) and RGC grant HKU 704611P.

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Ng, TW., Wu, CF. Nonlinear Loewy factorizable algebraic ODEs and Hayman’s conjecture. Isr. J. Math. 229, 1–38 (2019). https://doi.org/10.1007/s11856-018-1791-0

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  • DOI: https://doi.org/10.1007/s11856-018-1791-0

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