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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.
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참고문헌 (20)
참고문헌 신청
S. B. Bank / 1982 / Meromorphic solutions of the Riccati differential equation / Ann. Acad. Sci. Fenn. Ser. A I Math 6(2):369~398
D. C. Barnett / 2007 / Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations / Proc. Roy. Soc. Edinburgh Sect. A 137(3):457~474
W. Bergweiler / 1999 / Meromorphic solutions of some functional equations / Methods Appl. Anal 6(4):617~618
W. Bergweiler / 2007 / Zeros of differences of meromorphic functions / Math. Proc. Cambridge Philos. Soc 142(1):133~147
Z. X. Chen / 2011 / On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations / Sci. China Math 54(10):2123~2133
D. C. Barnett / 2007 / Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations / Proc. Roy. Soc. Edinburgh Sect. A 137(3):457~474
W. Bergweiler / 1999 / Meromorphic solutions of some functional equations / Methods Appl. Anal 6(4):617~618
W. Bergweiler / 2007 / Zeros of differences of meromorphic functions / Math. Proc. Cambridge Philos. Soc 142(1):133~147
Z. X. Chen / 2011 / On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations / Sci. China Math 54(10):2123~2133
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