On Holonomic Systems of Micro - differential Equations . Ill — Systems with Regular Singularities —

This is the third of the series of the papers dealing with holonomic systems(*. A holonomic system is, by definition, a left coherent (f-Module (or ^-Modules)* whose characteristic variety is Lagrangian. It shares the finiteness theorem with a linear ordinary differential equation, namely, all the cohomology groups associated with its solution sheaf are finite dimensional ([6], [12]). Hence the study of such a system will give us almost complete information concerning the functions which satisfy the system, as in the onedimensional case. Actually, analyzing special functions by the aid of the theory of ordinary differential equations is one of the most important subjects in the classical analysis. From this point of view, the study of holonomic systems with regular singularities is most important. However, even though the theory of linear ordinary differential equations with regular singularities has been developed quite successfully, the general theory of holonomic systems with regular singularities was not fully developed in the past, especially compared with the fruitful success attained in the one-dimensional case. Still it should be worth doing, and we hope we have established a solid basis for the theory in this paper. For example, we establish several basic results needed for the manipulation of holonomic systems with regular singularities, such as the integration and the restriction of such systems (Chapter V). We also give an analytic character-