The Stokes structure in asymptotic analysis 1: Bessel, Weber and hypergeometric functions

This is the first in a series of four papers entitled 'The Stokes structure in asymptotic analysis'. They introduce a methodology for the asymptotic analysis of differential equations with polynomial coefficients which also provides a further insight into the Stokes' Phenomenon. This approach consists of a chain of steps based on the concept of the Stokes Structure an algebraic-analytic structure, the idea of which emerges naturally from the monodromic properties of the Gauss hypergeometric function, and which can be treated independently of the differential equations, and Fourier-like transforms adjusted to this Stokes Structure. Every step of this approach, together with all its exigencies, is illustrated by means of the non-trivial treatment of Bessel's and Weber's differential equations. It will be the aim of our future series of papers to extend this approach to matrix differential equations. It is our great pleasure to publish this series of papers in our home town and to dedicate it to the memory of our dear teacher, Naum Il'ich Akhiezer, who taught us the basic knowledge of the theory of transcendental functions and inculcated in us the taste and the love for this theory.