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Some properties of solutions of certain complex linear differential equations with meromorphic coefficients. (English) Zbl 1185.34132
Consider the second-order linear differential equation
$f''+ A_1(z)\cdot e^{P(z)}\cdot f'+ A_0(z)\cdot e^{Q(z)}\cdot f= 0,\tag{1}$ where $$P(z)$$, $$Q(z)$$ are nonconstant polynomials, $$A_1(z)$$, $$A_0(z)$$ are entire nontrivial functions such that $$\rho(A_1)< \deg P(z)$$, $$\rho(A_0)< \deg Q(z)$$. G. Gundersen showed that if $$\deg P(z)\neq\deg Q(z)$$, then every nonconstant solution of (1) is of infinite order. If $$\deg P(z)= \deg Q(z)$$, then (1) may have nonconstant solution of finite order. For instance $$f(z)= e^z+ 1$$ satisfies $$f''+ e^z\cdot f'- c^z\cdot f= 0$$.
In the present paper, the authors study the relations between the solutions, their first and second derivatives of the differential equation (1) and meromorphic functions of finite order. Let $$\lambda(1/f)$$ denote the exponents of convergence of the pole-sequence of a meromorphic function $$f$$, let $$\overline\lambda(f)$$ to denote respectively the exponents of convergence of the sequence of distinct zeros of $$f$$.
The authors prove the following theorems:
Theorem 1.1. Let $$P(z)= \sum^n_{k=0} a_kz^k$$ and $$Q(z)= \sum^n_{k=0} b_k z^k$$ be nonconstant polynomials where $$a_k$$, $$b_k$$ are complex numbers, $$a_n\neq 0$$, $$b_n\neq 0$$ such that $$a_n= c\cdot b_n$$ $$(c>1)$$ and $$\deg(P-c\cdot Q)= m\geq 1$$ and $$A_1(z)$$, $$A_0(z)$$ be meromorphic nontrivial functions with $$\rho(A_j)< m$$. Let $$d_0(z)$$, $$d_1(z)$$, $$d_2(z)$$ be polynomials that are not all equal to zero, $$\varphi(z)$$ is a meromorphic nontrivial function with finite order. If $$f$$ is a meromorphic nontrivial solutions of (1) with $$\lambda(1/f)<\infty$$, then the differential polynomial $$g(z)= d_2f''+ d_1 f'+ d_0 f$$ satisfies $$\overline\lambda(g-\varphi)=\infty$$.
Theorem 1.2. Suppose that $$P(z)$$, $$Q(z)$$, $$A_1(z)$$, $$A_0(z)$$ satisfy the hypotheses of Theorem 1.1. If $$\varphi(z)$$ is a meromorphic nontrivial function with finite order then every meromorphic solution $$f$$ of equation (1) satisfies $$\overline\lambda(f-\varphi)= \overline\lambda(f'- \varphi)= \overline\lambda(f''- \varphi)= \infty$$.
##### MSC:
 34M03 Linear ordinary differential equations and systems in the complex domain 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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