On the extension of analytic solutions of a class of first-order q-difference equations

TL;DR

The paper investigates the existence and meromorphic continuation of solutions to a first-order $q$-difference equation of rational form with meromorphic coefficients: $y(qz)=\frac{a_1(z)y(z)+\cdots+a_p(z)y(z)^p}{1+b_1(z)y(z)+\cdots+b_t(z)y(z)^t}$. It introduces a constructive Banach fixed point framework to obtain existence and uniqueness of solutions in large-domain regions $D(\rho)$ and rectangles $D(\rho,\sigma)$, avoiding Nevanlinna-type growth conditions. Under explicit bounds on the coefficients and appropriate range for $|q|$, the results guarantee a unique meromorphic solution that extends to all of $\mathbb{C}$, even when the leading coefficient $a_1(z)$ vanishes. The work extends prior studies of complex difference equations by providing an analytic, fixed-point-based method for obtaining global meromorphic solutions to $q$-difference equations with meromorphic coefficients.

Abstract

In this paper, we use the Banach fixed point theorem to examine the existence of meromorphic solutions to the following first-order $q$-difference equation \begin{align}\tag{†}\label{dagger} y(qz)=\frac{a_1(z)y(z)+a_2(z)y(z)^2+\dots+a_p(z)y(z)^p}{1+b_1(z)y(z)+\cdots +b_t(z)y(z)^t}, \end{align} where $q\in \mathbb{C},$ $a_1(z), \dots, a_p(z); b_1(z), \dots, b_t(z)$ are all meromorphic functions. We establish sufficient conditions ensuring the existence and uniqueness of meromorphic solutions that can be extended to the entire complex plane $\mathbb{C}.$ More precisely, we have the following result. If $\left | q \right |\geq 3 $ and \[|a_1(z)| = \max_{1 \le j \le p} |a_j(z)| \le \frac{1}{|z|}, \quad \max_{1 \le k \le t} |b_k(z)| \le \frac{1}{|z|}, \quad z \in \{\, |\Re(z)| \ge ρ> 0 \,\}, \] and $y(0)\ne \infty,$ then we prove that~\eqref{dagger} admits a unique meromorphic solution in $D(ρ),$ which can be extended meromorphically to $\mathbb {C}.$ Moreover, if $a_1(z)\equiv 0,$ the conclusion still holds. Furthermore, if $\left | q \right |\geq 6$ and \begin{gather*} |a_1(z)| \le \frac{1}{|q|}, \quad |a_j(z)| \le |q|^{|z|} \quad (2 \le j \le p), \quad |b_k(z)| \le |q|^{|z|} \quad (1 \le k \le t), \\[4pt] z \in D(ρ,σ) = \{\, z : |\Re(z)| \le ρ,\; |\Im(z)| \le σ, \,\, ρ>0,\,\, σ>0 \,\}, \end{gather*} and $y(0)\ne \infty,$ then we prove that \eqref{dagger} admits a unique meromorphic solution in $D(ρ, σ),$ which can also be extended meromorphically to $\mathbb {C}.$ This conclusion remains valid in the case where $a_1(z)\equiv 0.$

Theorems & Definitions (11)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1: Banach fixed point theorem
• proof
• proof
• Lemma 5.1
• proof