Uniqueness of meromorphic solutions of the difference equation R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z)

Abstract

This paper mainly concerns the uniqueness of meromorphic solutions of first order linear difference equations of the form \t\t\t*R1(z)f(z+1)+R2(z)f(z)=R3(z),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z), $$\\end{document} where R_{1}(z)not equiv 0, R_{2}(z), R_{3}(z) are rational functions. Our results indicate that the finite order transcendental meromorphic solution of equation (*) is mainly determined by its zeros and poles except for some special cases. Examples for the sharpness of our results are also given.