On the extension of analytic solutions of first-order difference equations

TL;DR

This paper investigates the extension of analytic solutions of first-order difference equations with meromorphic coefficients, establishing conditions for global meromorphic solutions and analyzing their complex structure, including branch points and natural boundaries.

Contribution

It proves the existence of global meromorphic solutions under growth conditions on coefficients and analyzes the complex structure of solutions for specific equations.

Findings

Existence of global meromorphic solutions under certain coefficient bounds.

Solutions often have algebraic branch points and complex Riemann surface structures.

Identification of solutions with natural boundaries as described by Mahler.

Abstract

We will consider first-order difference equations of the form \[ y(z+1) = \frac{\lambda y(z)+a_2(z)y(z)^2+\cdots+a_p(z)y(z)^p}{1 + b_1(z)y(z)+\cdots+b_q(z)y(z)^q}, \] where $\lambda\in\mathbb{C}\setminus\{0\}$ and the coefficients $a_j(z)$ and $b_k(z)$ are meromorphic. When existence of an analytic solution can be proved for large negative values of $\Re(z)$, the equation determines a unique extension to a global meromorphic solution. In this paper we prove the existence of non-constant meromorphic solutions when the coefficients satisfy $|a_{j}(z)|\leq \nu^{|z|}$ and $|b_{k}(z)|\leq \nu^{|z|}$ for some $\nu<|\lambda|$ in a half-plane. Furthermore, when a solution exists that is analytic for large positive values of $\Re(z)$, the equation determines a unique extension to a global solution that will generically have algebraic branch points. We analyse a particular constant coefficient equation, $y(z+1)=\lambda y(z)+y(z)^2$, $0<\lambda<1$, and describe in detail the infinitely-sheeted Riemann surface for such a solution. We also describe solutions with natural boundaries found by Mahler.