On Uniqueness of Meromorphic Solutions to Some Differential-Difference Equations

In this paper, we prove the uniqueness of finite-order transcendental meromorphic solutions of differential-difference Painlevé III and V equations: ω(z+1)ω(z-1)+a(z)ω^(z)ω(z)=∑ₘ₌₀³aₘωᵐ(z)∑ₙ₌₀²bₙωⁿ(z), and (ω(z)ω(z+1)-1)(ω(z)ω(z-1)-1)+a(z)ω^(z)ω(z)=∑ₘ₌₀⁶aₘωᵐ(z)ω(z)-b₁(z), where aₘ , bₙ and $$a(z)$$ are small functions of solution $$f(z)$$ . We show that if the solution $$f(z)$$ shares e₁ , e₂ , and ∞ CM with another meromorphic function $$g(z)$$ , then f(z)≡ 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 e₁ and e₂ CM. MSC2020 numbers:30D35.