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)f'(z)/f(z)=R(z,f)=Σ_(m=0)~3 a_mf~m (z)/Σ_(n=0)~2 b_nf~n(z) and f(z+1)f(z-1)+a(z)f'(z)/f(z)=R(z,f)=Σ_(m=0)~4 a_mf~m(z)/Σ_(n=0)~3 b_nf~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≠e_2.
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