Zero order meromorphic solutions of $q$-difference equations of Malmquist type

TL;DR

This paper investigates zero order meromorphic solutions of first order $q$-difference equations of Malmquist type, showing reductions to linear or Riccati forms and providing explicit solutions in the autonomous case, with growth analysis of related composite functions.

Contribution

It characterizes conditions under which zero order meromorphic solutions exist for $q$-difference equations and connects these to linear, Riccati, and transformed equations, extending the understanding of their solution structure.

Findings

Solutions reduce to linear or Riccati equations when $|q| ot=1$.

Explicit solutions are provided for autonomous cases.

Growth properties of composite functions are analyzed in relation to the difference Painlevé property.

Abstract

We consider the first order $q$-difference equation \begin{equation}\tag{\dag} f(qz)^n=R(z,f), \end{equation} where $q\not=0,1$ is a constant and $R(z,f)$ is rational in both arguments. When $|q|\not=1$, we show that, if $(\dag)$ has a zero order transcendental meromorphic solution, then $(\dag)$ reduces to a $q$-difference linear or Riccati equation, or to an equation that can be transformed to a $q$-difference Riccati equation. In the autonomous case, explicit meromorphic solutions of $(\dag)$ are presented. Given that $(\dag)$ can be transformed into a difference equation, we proceed to discuss the growth of the composite function $f(\omega(z))$, where $\omega(z)$ is an entire function satisfying $\omega(z+1)=q\omega(z)$, and demonstrate how the proposed difference Painlev\'e property, as discussed in the literature, applies for $q$-difference equations.