On limiting directions of entire solutions of complex differential-difference equations

Abstract

In this article, we mainly obtain the measure of Julia limiting directions and transcendental directions of Jackson difference operators of non-trivial transcendental entire solutions for differential-difference equation \({f^n}(z) + \sum\limits_{k = 0}^n {{a_{{\lambda _k}}}(z){p_{{\lambda _k}}}(z,f) = h(z),} \) where \({p_{{\lambda _k}}}(z,f)\,\,\,(\lambda \in \mathbb{N})\) are distinct differential-difference monomials, \({a_{{\lambda _k}}}(z)\) are entire functions of growth smaller than that of the transcendental entire h(z). For non-trivial entire solutions f of differential-difference equation \({P_2}(z,f) + {A_1}(z){P_1}(z,f) + {A_0}(z) = 0,\) where P_λ(z,f)(λ = 1, 2) are differential-difference polynomials. By considering the entire coefficient associated with Petrenko’s deviation, the measure of common transcendental directions of classical difference operators and Jackson difference operators of f was studied.