Theory of hypergeometric functions

✍ Scribed by Kazuhiko Aomoto, Michitake Kita (auth.)

• Publisher: Springer Tokyo
• Year: 2011
• Tongue: English
• Leaves: 335
• Series: Springer Monographs in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xvi
Introduction: the Euler−Gauss Hypergeometric Function....Pages 1-19
Representation of Complex Integrals and Twisted de Rham Cohomologies....Pages 21-101
Arrangement of Hyperplanes and Hypergeometric Functions over Grassmannians....Pages 103-182
Holonomic Difference Equations and Asymptotic Expansion....Pages 183-259
Back Matter....Pages 261-317

✦ Subjects

Geometry; Functional Analysis