The Analytic Solutions of the Functional Equations a1f(γ1s)+a2f(γ2s)+⋯+anf(γns)=h(s).

In this paper, a sufficient and necessary condition of the existence of analytic solutions of the functional equation a 1 f (γ 1 s) + a 2 f (γ 2 s) + ⋯ + a n f (γ n s) = h (s) for s ∈ S is presented, where S ⊂ C is a given domain and h is a polynomial. This result generalizes some known results. A similar theorem on the meromorphic solutions of the equation is obtained. We also describe the structure of the set of solutions of the equation. Moreover, we give a result on the d-homogeneous solution of the homogeneous equation of the above equation for the high-dimensional case. [ABSTRACT FROM AUTHOR]