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

Abstract

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