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

TL;DR

This paper establishes conditions under which solutions to a class of first-order q-difference equations exist uniquely and can be extended meromorphically across the entire complex plane, using the Banach fixed point theorem.

Contribution

It provides new sufficient conditions for the existence and uniqueness of meromorphic solutions to first-order q-difference equations, extending solutions globally under specific bounds.

Findings

Unique meromorphic solutions exist under certain bounds on coefficients.

Solutions can be extended meromorphically to the entire complex plane.

Results apply to equations with or without the leading coefficient a_1(z).

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{{\dag}}\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 \[|a_1(z)| = \max_{1 \le j \le p} |a_j(z)| \le \frac{1}{|z|}, \quad \max_{1 \le k \le t} |b_k(z)| \le \frac{1}{|z|}, \quad z \in \{\, |\Re(z)| \ge \rho > 0 \,\}, \] and $y(0)\ne \infty,$ then we prove that~\eqref{dagger} admits a unique meromorphic solution in $D(\rho),$ 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(\rho,\sigma) = \{\, z : |\Re(z)| \le \rho,\; |\Im(z)| \le \sigma, \,\, \rho>0,\,\, \sigma>0 \,\}, \end{gather*} and $y(0)\ne \infty,$ then we prove that \eqref{dagger} admits a unique meromorphic solution in $D(\rho, \sigma),$ which can also be extended meromorphically to $\mathbb {C}.$ This conclusion remains valid in the case where $a_1(z)\equiv 0.$