Aomoto, Kazuhiko.
Theory of Hypergeometric Functions [electronic resource] / by Kazuhiko Aomoto, Michitake Kita. - XVI, 320 p. online resource. - Springer Monographs in Mathematics, 1439-7382 . - Springer Monographs in Mathematics, .
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.
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.
9784431539384
10.1007/978-4-431-53938-4 doi
Mathematics.
Functional analysis.
Geometry.
Mathematics.
Geometry.
Functional Analysis.
QA440-699
516
Theory of Hypergeometric Functions [electronic resource] / by Kazuhiko Aomoto, Michitake Kita. - XVI, 320 p. online resource. - Springer Monographs in Mathematics, 1439-7382 . - Springer Monographs in Mathematics, .
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.
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.
9784431539384
10.1007/978-4-431-53938-4 doi
Mathematics.
Functional analysis.
Geometry.
Mathematics.
Geometry.
Functional Analysis.
QA440-699
516