Lower Bound Estimates of the Order of Meromorphic Solutions to Non-Homogeneous Linear Differential-Difference Equations

TL;DR

This paper establishes lower bound estimates for the growth order of meromorphic solutions to non-homogeneous linear differential-difference equations with entire or meromorphic coefficients, under specific conditions on the coefficients.

Contribution

It provides new lower bound estimates for the order of solutions based on the dominant coefficient's growth characteristics.

Findings

Derived lower bounds for the order of solutions.

Identified conditions on coefficients that influence solution growth.

Extended understanding of solution behavior in differential-difference equations.

Abstract

In this article, we deal with the order of growth of solutions of non-homogeneous linear differential-difference equation \begin{equation*} \sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=F(z), \end{equation*} where $A_{ij},$ $F\left( z\right) $ are entire or meromorphic functions and $c_{i}$ $\left( 0,1,...,n\right) $ are non-zero distinct complex numbers. Under the sufficient condition that there exists one coefficient having the maximal lower order or having the maximal lower type strictly greater than the order or the type of other coefficients, we obtain estimates of the lower bound of the order of meromorphic solutions of the above equation.