On Meromorphic Solutions of Non-linear Differential-Difference Equations

Abstract

In this paper, we investigate the non-existence of transcendental entire solutions for non-linear differential-difference equations of the forms fn(z)+Q(z,f)=β1eα1z+β2eα2z+⋯+βseαsz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f^{n}(z)+Q(z,f)=\\beta _{1}e^{\\alpha _{1}z}+\\beta _{2}e^{\\alpha _{2}z}+\\cdots +\\beta _{s}e^{\\alpha _{s}z} \\end{aligned}$$\\end{document}and fn(z)f(k)(z)+Ld(z,f)=∑i=1spi(z)eαi(z),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f^{n}(z)f^{(k)}(z)+L_d(z,f)=\\sum ^{s}_{i=1}p_i(z)e^{\\alpha _i{(z)}}, \\end{aligned}$$\\end{document}where n, s are positive integers, nge s+2,Q(z, f) is a differential-difference polynomial in f of degree d.