Special Function of Several Variables

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1. On the study of (k, n) type hypergeometric functions (We abbreviate hypergeomentric as HF) : (1) The theory of twisted homologies and twisted cohomologies and interesection numbers was developed. We proposed a general framework based on the algebraic topology of the configulation spaces of n-points. A twisted simplicial theory and a twisted singular theory for the configulation spaces was developped and the exterior power structure of the homology module associated with the HG system of (3,6) type was obtained and all the cases where the group is of frute order were determined. (3) The relation between contiguous operators operators and the b functions for HG functions were obtained.2. On the study of(k, n)lambda type HG functions where lambda=(lambda_1, ・・・ lambda_1) is the decomposition of n : (1) The structure of the Lie algebra generated by contiguous operators for (k, n)lambda type HG functions was determined. (2) We found a limit process which reduces (k, n)lambda type HG functions (or system) to (k, n)mu type HG functions (or system) where mu is adjacent to lamdba. The process is just the confluence from Gauss to Kummer in the case where k = 2, n = 4 and lambda = (1,1,1,1), mu = (2,1,1,1). It acts well on Jodan groups, on characters and on differential equations.3. On asymptotic analysis : A general methpd of asymptotic for confluent HG functions was proposed. It consists of three parts : the first is Borel Transformation, the second is an analysis of the resurgent equation and the last is Laplace transformation. Based on the method, Stokes multipliers for the confluent HG two variables were calculated.