Meromorphic solutions of algebraic difference equations

TL;DR

This paper investigates meromorphic solutions of a specific algebraic difference equation, showing their connection to differential equations, algebraic relations between solutions, and similar growth behaviors under certain conditions.

Contribution

It establishes the existence of a continuous limit from the difference equation to a differential equation and analyzes the properties of solutions, including algebraic relations and growth behaviors.

Findings

Solutions have a continuous limit to the differential equation.

Two distinct transcendental solutions satisfy an algebraic relation.

Solutions exhibit similar growth behaviors under certain conditions.

Abstract

It is shown that the difference equation \begin{equation}\label{abseq} (\Delta f(z))^2=A(z)(f(z)f(z+1)-B(z)), \qquad\qquad (1) \end{equation} where $A(z)$ and $B(z)$ are meromorphic functions, possesses a continuous limit to the differential equation \begin{equation}\label{abseq2} (w')^2=A(z)(w^2-1),\qquad\qquad (2) \end{equation} which extends to solutions in certain cases. In addition, if (1) possesses two distinct transcendental meromorphic solutions, it is shown that these solutions satisfy an algebraic relation, and that their growth behaviors are almost same in the sense of Nevanlinna under some conditions. Examples are given to discuss the sharpness of the results obtained. These properties are counterparts of the corresponding results on the algebraic differential equation (2).