Properties on $q$-difference Riccati equation
Bull. Korean Math. Soc. 2018 Vol. 55, No. 6, 1755-1771
Published online May 2, 2018
Printed 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
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