On meromorphic solutions of the systems of linear differential equations with meromorphic coefficients (Q5918498)

Let \(M\) denote a field of meromorphic functions in \(\mathbf{C}\). This paper considers systems of linear differential equations whose dimensionality can be reduced and which have the form \[ w_i'=\sum_{k=1}^na_{ik} w_k \quad (i=1,\dots,n) \tag{1} \] where the \(a_{ik}\) are in \(M\). Estimates are obtained for the growth rate of meromorphic vector solutions to (1) without any additional constraints on the order of growth of the \(a_{ik}\). The results are based on the reduction of dimensionality of the system through use of the corresponding transformation to that employed by \textit{W. Hengartner} [Comment. Math. Helv. 42, 60--80 (1967; Zbl 0173.33704)] for his study of properties of the vector solutions of systems of linear differential equations where the coefficients \(a_{ik}\) and the components of the vector solutions are entire functions.