On \(q\)-difference Riccati equations and second-order linear \(q\)-difference equations (Q355894)

Summary: We consider \(q\)-difference Riccati equations and second-order linear \(q\)-difference equations in the complex plane. We present some basic properties, such as the transformations between these two equations, the representations and the value distribution of meromorphic solutions of \(q\)-difference Riccati equations, and the \(q\)-Casorati determinant of meromorphic solutions of second-order linear \(q\)-difference equations. In particular, we find that the meromorphic solutions of these two equations are concerned with the \(q\)-gamma function when \(q \in \mathbb C\) such that \(0 < |q| < 1\). Some examples are also listed to illustrate our results.