Asymptotics solutions of equations with higher-order degeneracies

Asymptotic decompositions of solutions to degenerate equations that include a differential operator with smooth coefficients are studied. Laplace-Borel transform is applied to homogeneous equations in the case where the roots of the operator symbol are simple. The space of holomorphic functions of exponential growth in the domain is considered and the space of entire functions of exponential growth is denoted. asymptotics for solutions of the homogeneous equation are constructed under the assumption that the spectral points are simple. If the function has poles of the first order at some points, then the asymptotics of the solution to the homogeneous equation has the form where the summation is over the union of all roots of the polynomial.