Study of solutions of logarithmic order to higher order linear differential-difference equations with coeffcients having the same logarithmic order

Abstract

Abstract. The main purpose of this paper is to study the growth of solutions of the linear differential-difference equationwhere Aij(z) (i = 0, · · · , n; j = 0, · · · , m) are entire or meromorphic functions of finite logarithmic order and ci (0, · · · , n) are distinct complex numbers. We extend some precedent results due to Wu and Zheng and others.1. Introduction and main results. Throughout this paper, we assume that readers are familiar with the standard notations and the fundamental results of the Nevanlinna value distribution theory of meromorphic functions ([12, 19]). Let f be a meromorphic function; we define m(r, f ) = 1 2πand T (r, f ) = m(r, f ) + N (r, f ) (r > 0)2010 Mathematics Subject Classification. 30D35, 34K06, 34K12.

Growth and value distribution of linear difference polynomials generated by meromorphic solutions of higher-order linear difference equations

“…Recently, there has been an increasing interest in the study on the properties of meromorphic solutions of complex difference equations from the viewpoint of difference analogues of Nevanlinna theory (see [4,6,8]) and among those many good results are obtained for the case of complex linear difference equations (see [4,5,6,11,12,14,16,19,20]). For the case of complex linear differential-difference equations see [1,2,3,13,15,21]. In particular, inspired by the results about the growth and the value distribution of differential polynomials generated by meromorphic solutions of complex linear differential equations, Latreuch and Belaïdi in [11] investigated the growth of linear difference polynomials generated by meromorphic solutions of the second order complex linear difference equation f (z + 2) + a(z)f (z + 1) + b(z)f (z) = 0, (1.1) where a(z) and b(z) are meromorphic functions.…”

Finite Logarithmic Order Meromorphic Solutions of Complex Linear Delay-Differential Equations

“…Recently, the research on the properties of meromorphic solutions of complex delay-differential equations has become a subject of great interest from the viewpoint of Nevanlinna theory and its difference analogues. In [20], Liu, Laine and Yang presented developments and new results on complex delay-differential equations, an area with important and interesting applications, which also gathers increasing attention (see, [4,5,8,24]. In [8], Chen and Zheng considered the following homogeneous complex delay-differential equation…”

Picturing the Growth Order of Solutions in Complex Linear Differential–Difference Equations with Coefficients of φ-Order

“…A number of authors (cf. [6,[26][27][28][29]) have studied the growth rate of any meromorphic solution of linear differential-difference equations defined by…”