Аннотация
A new approach to the study of periodic by all independent variables system of equations with a differentiation operator solutions along the direction of the main diagonal and with delayed arguments is proposed. The essence of the approach is to reduce the study of the multi-periodic solution of a linear inhomogeneous system to the construction of a solution of a simpler linear differential-difference system on the basis of the method of variating arbitrary constants of the complete integral of a homogeneous system. An integral representation of the unique multiperiodic solution of an inhomogeneous system is presented, expressed by a functional series of terms given by multiple repeated integrals. An estimate is given for the norm of a multi-periodic solution.