n阶复线性微分方程的无穷φ-级解
Infinite φ-Order Solutions of n-Order Complex Linear Differential Equations
摘要: 本文主要研究以下
n阶复线性微分方程
f(n) +
An−1(
z)
f(n−1) + ··· +
A1(
z)
f′ +
A0(
z)
f =
F(
z)，其中系数
A0,
A1, ...,
An−1,
F(
z)均为整函数且
A0(
z) ≢ 0。基于
φ-级增长度量给出系数需要满足的判定条件，证明该方程所有非常数整函数解均具有无穷
φ-级，并建立了解的超
φ-级与方程主导系数
φ-级之间的关系，推广了二阶复线性微分方程的已有相关结果。
Abstract: This paper mainly studies the following n-order complex linear differential equation f(n) + An−1(z)f(n−1) + ··· + A1(z)f′ + A0(z)f = F(z), where the coefficients A0, A1, ..., An−1, F(z) are all entire functions and A0(z) ≢ 0. We find some appropriate conditions on the A0, A1, ..., An−1, F(z) in terms of the φ-order growth metric which guarantee that every non-constant entire solution f of the equation have infinite φ-order, along with an additional relationship between the hyper φ-order of f and the φ-order of dominating coefficient in the equation, this generalizes the already relevant results of the second order complex linear differential equation.
参考文献
|
[1]
|
Yu, H. and Li, X. (2025) On the Infinite φ-Order Solutions of Second Order Linear Differential Equations. Computational Methods and Function Theory, 25, 645-662.
https://doi.org/10.1007/s40315-024-00548-1
|
|
[2]
|
Heittokangas, J., Wang, J., Wen, Z.T. and Yu, H. (2021) Meromorphic Functions of Finite φ-Order and Linear q-Difference Equations. Journal of Difference Equations and Applications, 27, 1280-1309.
https://doi.org/10.1080/10236198.2021.1982919
|
|
[3]
|
Zheng, J.H. (2010) Value Distribution of Meromorphic Functions. Springer.
https://doi.org/10.1007/978-3-642-12909-4
|
|
[4]
|
Hayman, W.K. (1964) Meromorphic Functions. Clarendon.
|
|
[5]
|
Gundersen, G.G. (1988) Finite Order Solutions of Second Order Linear Differential Equations. Transactions of the American Mathematical Society, 305, 415-429.
https://doi.org/10.2307/2001061
|
|
[6]
|
Chen, Z.X. and Yang, C.C. (1999) Some Further Results on the Zeros and Growths of Entire Solutions of Second Order Linear Differential Equations. Kodai Mathematical Journal, 22, 273-285.
https://doi.org/10.2996/kmj/1138044047
|
|
[7]
|
Laine, I. (1993) Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter.
https://doi.org/10.1515/9783110863147
|
|
[8]
|
Heittokangas, J., Yu, H. and Zemirni, M.A. (2021) On the Number of Linearly Independent Admissible Solutions to Linear Differential and Linear Difference Equations. Canadian Journal of Mathematics, 73, 1556-1591.
https://doi.org/10.4153/s0008414x20000607
|
|
[9]
|
Yi, H.X. and Yang, C.C. (2003) Uniqueness Theory of Meromorphic Functions. Kluwer.
https://doi.org/10.1007/978-94-017-3626-8
|