Bulletin of the
Korean Mathematical Society BKMS

In this paper, we investigate a certain type of $q$-difference Riccati equation in the complex plane. We prove that $q$-difference Riccati equation possesses a one parameter family of meromorphic solutions if it has three distinct meromorphic solutions. Furthermore, we find that all meromorphic solutions of $q$-difference Riccati equation and corresponding second order linear $q$-difference equation can be expressed by $q$-gamma function if this $q$-difference Riccati equation admits two distinct rational solutions and $q\in\mathbb{C}$ such that $0<|q|<1$. The growth and value distribution of differences of meromorphic solutions of $q$-difference Riccati equation are also treated.

Keywords: $q$-difference Riccati equation, $q$-difference equation, $q$-gamma function

MSC numbers: Primary 39B32; Secondary 30D35