On the complete description of the Stokes geometry for the first Painlevé hierarchy

with our sincerest congratulations on his being awarded Prix de l’Etat (Academie des Science). One of the central issues of this article is the introduction of the notion of virtual turning points for higher order Painlevé equations, and two of the authors (Kawai and Takei), together with T. Aoki, fondly remember the stimulating and comfortable conference (Algebraic analysis of singular perturbations, 1991), which Professor Boutet de Monvel, together with Professor M. Sato, organized, and where the notion of a virtual turning point for linear ordinary differential equations was first made public (under the modest name “a new turning point”). The notion of virtual turning points is one of the most important gifts to the exact WKB analysis from microlocal analysis, and hence we believe this article to be most appropriate to dedicate to Professor Boutet de Monvel, who has made substantial contributions to the development of microlocal analysis and asymptotic analysis.