Malmquist-type Results for Difference Equations with Periodic Coefficients

Abstract

An earlier result by N. Yanagihara leads us to consider the nature of a meromorphic solution $$f$$ of a difference equation $$\begin{aligned} f(z+c)=\sum _{j=0}^n p_j(z)(f(z))^j, \end{aligned}$$ where $$p_j$$ are periodic entire functions of the period $$c\in \mathbb {C}$$ , $$p_n \not \equiv 0$$ and $$n >1$$ . We shall show that if $$f$$ is non-periodic entire and of finite order of growth, it must be algebraic over any field that contains the coefficients of the difference equation. We also consider the special case $$n=2$$ more carefully and obtain specific information on the solution $$f$$ . Our methods are based on Nevanlinna theory and algebraic field theory.