Abstract
In this paper, we consider the q-difference equation \t\t\t(f(qz)+f(z))(f(z)+f(z/q))=R(z,f),\\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}$$ \\bigl(f(qz)+f(z)\\bigr) \\bigl(f(z)+f(z/q)\\bigr)=R(z,f), $$\\end{document} where R(z,f) is rational in f and meromorphic in z. It shows that if the above equation assumes an admissible zero-order meromorphic solution f(z), then either f(z) is a solution of a q-difference Riccati equation or the coefficients satisfy some conditions.
Highlights
1 Introduction In this paper, we use the basic notions of Nevanlinna theory [1,2,3,4] such as the characteristic function T(r, f ), counting function N(r, f ), and proximity function m(r, f )
A meromorphic solution f of a difference equation is called admissible if all the coefficients of the equation are in S(f )
An ordinary differential equation is said to possess the Painlevé property if all of its solutions are single-valued about all movable singularities, see [5]
Summary
We use the basic notions of Nevanlinna theory [1,2,3,4] such as the characteristic function T(r, f ), counting function N(r, f ), and proximity function m(r, f ). Abstract In this paper, we consider the q-difference equation (f (qz) + f (z))(f (z) + f (z/q)) = R(z, f ), where R(z, f ) is rational in f and meromorphic in z. It shows that if the above equation assumes an admissible zero-order meromorphic solution f (z), either f (z) is a solution of a q-difference Riccati equation or the coefficients satisfy some conditions.
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