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

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.