Theory of Hypergeometric Functions - Aomoto, Kazuhiko; Kita, Michitake; SpringerLink (Online service) (Tokyo : Springer Japan : Imprint: Springer, 2011)
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$1 (Bs rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff$1 (Bs classical theory on analytic difference equations on the other.s
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.
Keywords: Mathematics; Functional analysis; Geometry; Mathematics; Geometry; Functional Analysis