On Uniqueness of Meromorphic Solutions to Delay Differential Equation

Abstract: In this paper, we investigate uniqueness of finite-order transcendental meromorphic solutions of the following two equations: $$f(z+1)-f(z-1)+a(z)\frac{f^{\prime}(z)}{f(z)}=R(z,f)=\frac{\sum_{m=0}^{3}a_{m}f^{m}(z)}{\sum_{n=0}^{2}b_{n}f^{n}(z)},$$ and $$f(z+1)f(z-1)+a(z)\frac{f^{\prime}(z)}{f(z)}=R(z,f)=\frac{\sum_{m=0}^{4}a_{m}f^{m}(z)}{\sum_{n=0}^{3}b_{n}f^{n}(z)},$$ where $$R(z,f)$$ is an irreducible rational function in $$f(z)$$ , $$a(z)$$ , $$a_{m}$$ and $$b_{n}$$ are small functions of $$f(z)$$ . Such solutions $$f(z)$$ are uniquely determined by their poles and the zeros of $$f(z)-e_{j}$$ (counting multiplicities) for two complex numbers $$e_{1}\neq e_{2}$$ .