Differential polynomials generated by some complex linear differential equations with meromorphic coefficients. (English) Zbl 1166.34054

Belaïdi, Benharrat; el Farissi, Abdallah

Glas. Mat., III. Ser. 43, No. 2, 363-373 (2008).

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 \(f\).

Reviewer: Nikolay Vasilye Grigorenko (Kyïv)

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

Keywords:

meromorphic solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros