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Differential polynomials generated by some complex linear differential equations with meromorphic coefficients. (English) Zbl 1166.34054
Let $$M$$ be the field of meromorphic functions in $$C$$. Consider the second order differential equation
$y''+A_{1}(z)\exp(P(z))y'+A_{0}(z)\exp(Q(z))y=0,\tag{1}$ where $$A_{1},\, A_{0}\in M,\, P,\, Q\in C[z]$$. The authors study properties of meromorphic solutions of equation (1) and prove the following theorem.
Theorem: Let $$P(z)=\sum_{i=0}^{n}a_{i}z^{i}$$ and $$Q(z)=\sum_{i=0}^{n}b_{i}z^{i}$$ be nonconstant polynomials, where $$a_{i},\, b_{i}\in C\,(i=0,\dots, n),a_{n}b_{n}\neq0$$ such that $$\arg a_{n}\neq\arg b_{n}$$ or $$a_{n}=cb_{n}(0<c<1)$$ and $$A_{1}(z),\, A_{0}(z)\,(\neq0)$$ be meromorphic functions with $$\rho(A_{j})<n\,(j=0,1)$$ . Let $$d_{0},d_{1},d_{2}\in M$$ that are not all equal to zero with $$\rho(d_{j})<n\,(j=0,1,2),\varphi\in M^{*}$$ has finite order. If $$f\in M^{*}$$ is a solution of (1), then the differential polynomial $$g=d_{2}f''+d_{1}f'+d_{0}f$$ satisfies $$\bar{\lambda}(g-\varphi)=\infty$$, where $$\bar{\lambda}(f)$$ denotes the exponents of covergence of the sequence of distinct zeros of $$f$$.

##### MSC:
 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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