Exponential Polynomials as Solutions of Differential-Difference Equations of Certain Types

We consider the exponential polynomials solutions of non-linear differential-difference equation f(z)n+q(z)eQ(z)f(k)(z+c)=P(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f(z)^{n}+q(z)e^{Q(z)}f^{(k)}(z+c) = P(z)}$$\end{document}, where q(z), Q(z), P(z) are polynomials and n, k are positive integers and the linear differential-difference equation f′(z)=f(z+c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f'(z) = f(z + c)}$$\end{document}. Our results show that any exponential polynomials’ solutions of the above two differential-difference equations should have special forms. This paper is a continuation of Wen et al. (Acta Math Sin 28(7):1295–1306, 2012).