Theory of Hypergeometric Functions [Paperback]

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-Delignes rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoffs classical theory on analytic difference equations on the other.

This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. It shows that hypergeometric integrals generally satisfy holonomic system of linear differential equations.

1 Introduction: the Euler-Gauss Hypergeometric Function.- 2 Representation of Complex Integrals and Twisted de Rham Cohomologies.- 3 Hypergeometric functions over Grassmannians.- 4 Holonomic Difference Equations and Asymptotic Expansion References Index.

Reader will understand clearly multidimensional hypergeometric function as a natural extension of the classical one from viewpoint of integrals

A 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 functions

Application of hypergeometric functions to several analytic or geometric problems