Abstract
In this paper, we prove the uniqueness of finite-order transcendental meromorphic solutions of differential-difference Painlevé III and V equations:
\(\omega(z+1)\omega(z-1)+a(z)\frac{\omega^{\prime}(z)}{\omega(z)}=\frac{\sum_{m=0}^{3}a_{m}\omega^{m}(z)}{\sum_{n=0}^{2}b_{n}\omega^{n}(z)},\)
and
\((\omega(z)\omega(z+1)-1)(\omega(z)\omega(z-1)-1)+a(z)\frac{\omega^{\prime}(z)}{\omega(z)}=\frac{\sum_{m=0}^{6}a_{m}\omega^{m}(z)}{\omega(z)-b_{1}(z)},\)
where \(a_{m}\) , \(b_{n}\) and \(a(z)\) are small functions of solution \(f(z)\) . We show that if the solution \(f(z)\) shares \(e_{1}\) , \(e_{2}\) , and \(\infty\) CM with another meromorphic function \(g(z)\) , then \(f(z)\equiv g(z)\) . Moreover, for Painlevé V equation, if \(g(z)\) is replaced by \(f(z+c)\) , it is sufficient for \(f(z)\) and \(f(z+c)\) to share the values \(e_{1}\) and \(e_{2}\) CM.
MSC2020 numbers:30D35.
