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

TL;DR

The paper addresses the extension problem for analytic solutions of first-order difference equations with meromorphic coefficients, establishing conditions under which analytic solutions extend to global meromorphic functions and when they develop algebraic branching. The authors provide a direct Banach fixed-point proof for the autonomous case with a fixed point of multiplier one, derive explicit asymptotic forms for the solutions, and show how non-autonomous equations with growth-bound coefficients admit unique global meromorphic (or analytic) solutions depending on the sign of $|oldsymbol{ m abla}|$ relative to 1. They also analyze a constant-coefficient model $y(z+1)=oldsymbol{ m abla} y(z)+ y(z)^2$ with $0<oldsymbol{ m abla}<1$, describing an infinitely-sheeted Riemann surface and connecting to Mahler's natural-boundary phenomena. The results thus illuminate the global analytic structure and branching of solutions, including infinite-sheeted covers and natural boundaries.

Abstract

We will consider first-order difference equations of the form \[ y(z+1) = \frac{λy(z)+a_2(z)y(z)^2+\cdots+a_p(z)y(z)^p}{1 + b_1(z)y(z)+\cdots+b_q(z)y(z)^q}, \] where $λ\in\mathbb{C}\setminus\{0\}$ 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 $|a_{j}(z)|\leq ν^{|z|}$ and $|b_{k}(z)|\leq ν^{|z|}$ for some $ν<|λ|$ 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, $y(z+1)=λy(z)+y(z)^2$, $0<λ<1$, and describe in detail the infinitely-sheeted Riemann surface for such a solution. We also describe solutions with natural boundaries found by Mahler.

Figures (2)

• Figure 1: Branch points at $\hat{r}$ and $\hat{r}/\lambda$
• Figure 2: Branch point at $\hat{r}/\lambda^2$

Theorems & Definitions (11)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• proof : Proof of Claim 1
• proof : Proof of Claim 2
• proof : Proof of Claim 3