关于差分Riccati方程解的存在性

On existence of solutions of difference Riccati equations

研究差分Riccati方程f(qz+c)=A(z)f(z)+B(z)/C(z)+D(z),其中A、B、C、D为亚纯函数,得到解簇为H(f(qz+c))=f 0(z),f(z)=(f 1(z)-f 0(z))(f 2(z)-f 0(z))/Q(z)(f 2(z)-f 1(z))+(f 2(z)-f 0(z))+f 0(z),这里Q(z)为任意的满足Q(z)=Q(qz+c)的亚纯函数,且f 0(z)、f 1(z)、f 2(z)为方程的3个互异的亚纯函数解。推广了Chen与Shon的最近结果。

In this paper,we mainly study on the existence of solutions of difference Riccati equation f(qz+c)=A(z)f(z)+B(z)/C(z)+D(z),where q,c are two nonzero complex numbers,A,B,C,D are meromorphic functions,and obtain that its solution family with one-parameter H(f(qz+c))={f0(z),f(z)=(f1(z)-f0(z))(f2(z)-f0(z))/Q(z)(f2(z)-f1(z))+(f2(z)-f0(z))+f0(z)},where Q(z)is any meromorphic function satisfying Q(z)=Q(qz+c),and f0(z),f1(z),f2(z)are three distinct meromorphic solutions.This is an extension of a recent result due to Chen and Shon.