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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 \(f\).

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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