Theory of Hypergeometric Functions
This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy a holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy a holonomic system of linear difference equations with respect to the exponents. These are deduced from Grothendieck-Deligne’s rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff’s classical theory on analytic difference equations on the other.
Reader will understand clearly multidimensional hypergeometric function as a natural extension of the classical one from viewpoint of integralsA quick introduction to rational de Rham cohomology due to A.Grothendieck and P.Deligne and also to holonomic differential equations (or Gauss-Manin connection) and difference equations associated with hypergeometric functionsApplication of hypergeometric functions to several analytic or geometric problems