Application of the Repeated Quantization Method to a Class of Non-Fuchsian Equations

One of the fundamental problems of the analytic theory of ordinary differential equations is the problem of constructing asymptotics of solutions of differential equations in the neighborhood of irregular singular points. In general, this problem has not yet been solved. However, in recent years, the method of repeated quantization has been created to solve this problem, which allows constructing asymptotics of solutions for a wide class of equations with irregular singularities. This work is devoted to the development of this method. For example, this method has been used to construct asymptotic solutions for differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point, which, generally speaking, is irregular. The method of repeated quantization is based on the methods of resurgent analysis, that is, on the application of the Laplace-Borel transform. This method is applied when the roots of the principal symbol are multiple. Using the results of this article, the class of equations with irregular singular points to which the repeated quantization method is applicable is expanded. Namely, to those equations with an irregular singular point such that the asymptotics of solutions of the original equation in the Laplace-Borel images contain exponentials in whose exponents there are polynomials of a fractional degree of the variable. The application of the obtained results to an equation of this type is illustrated by a specific example.