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

Abstract

We consider the first order $q$-difference equation \begin{equation}\tag†f(qz)^n=R(z,f), \end{equation} where $q\ne 0,1$ is a constant and $R(z,f)$ is rational in both arguments. When $|q|\ne 1$, we show that, if $(\hat{a}€)$ has a zero order transcendental meromorphic solution, then $(\hat{a}€)$ 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 $(\hat{a}€)$ are presented. Given that $(\hat{a}€)$ can be transformed into a difference equation, we proceed to discuss the growth of the composite function $f(\ddot{I}‰(z))$, where $\ddot{I}‰(z)$ is an entire function satisfying $\ddot{I}‰(z+1)=q\ddot{I}‰(z)$, and demonstrate how the proposed difference Painlevé property, as discussed in the literature, applies for $q$-difference equations.

13 pages