On meromorphic solutions of some linear differential equations with entire coefficients being Fabry gap series

摘要

In this paper, we investigate the growth and the exponent of convergence of the sequence of φ-points of meromorphic solutions of the linear differential equations A k ( z ) f ( k ) + A k ? 1 ( z ) f ( k ? 1 ) + ? + A 1 ( z ) f ′ + A 0 ( z ) f = 0 $$A_{k}(z)f^{(k)}+A_{k-1}(z)f^{(k-1)}+ cdots+A_{1}(z)f'+A_{0}(z)f=0 $$ and A k ( z ) f ( k ) + A k ? 1 ( z ) f ( k ? 1 ) + ? + A 1 ( z ) f ′ + A 0 ( z ) f = F ( z ) , $$A_{k}(z)f^{(k)}+A_{k-1}(z)f^{(k-1)}+ cdots+A_{1}(z)f'+A_{0}(z)f=F(z), $$ with entire coefficients A j ( z ) $A_{j}(z)$ , j = 0 , 1 , … , k $j=0,1,ldots,k$ and F ( z ) $F(z)$ , where k ≥ 2 $kgeq2$ , A 0 ( z ) A k ( z ) ? 0 $A_{0}(z)A_{k}(z)otequiv0$ , φ ( z ) $arphi(z)$ is a meromorphic function of finite order, and there is only one dominant coefficient A k ( z ) $A_{k}(z)$ of the maximal order, which is also a Fabry gap series.