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 Properties on $q$-difference Riccati equation Bull. Korean Math. Soc. 2018 Vol. 55, No. 6, 1755-1771 https://doi.org/10.4134/BKMS.b171049Published online November 30, 2018 Zhi-Bo Huang, Ran-Ran Zhang South China Normal University, Guangdong University of Education Abstract : 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 Downloads: Full-text PDF