Theory of hypergeometric functions. With an appendix by Toshitake Kohno.

*(English)*Zbl 1229.33001
Springer Monographs in Mathematics. Berlin: Springer (ISBN 978-4-431-53912-4/hbk; 978-4-431-53938-4/ebook). xvi, 317 p. (2011).

This book is an English translation (by Kenji Iohara) of the original Japanese edition by the two authors Kazuhiko Aomoto and Michitake Kita (deceased in 1995). Furthermore, Appendix D on the KZ equation is written by Toshitake Kohno.

The book contains four chapters and four appendices. The first chapter is a short (19 pages) introduction of the Euler-Gauss hypergeometric function and its basic properties. The second chapter (80 pages) deals with “Representation of complex integrals and twisted de Rham cohomologies”. The third chapter (80 pages) deals with “Arrangement of hyperplanes and hypergeometric functions over Grassmannians” and the fourth chapter (77 pages) with “Holonomic difference equations and asymptotic expansion”. Each chapter is divided in numerous sections and subsections.

There are four appendices on “Mellin’s generalized hypergeometric functions”, “The Selberg integral and hypergeometric functions of BC type”, “Monodromy representation of hypergeometric functions of type \((2,m+1;\alpha)\)” and the already mentioned Appendix D.

The book ends with eight pages of references and three index pages. The text of the book is not an exhaustive textbook treatment of special functions. However, the authors construct – from the viewpoint that hypergeometric functions are complex integrals of complex powers of polynomials – a unified theory of hypergeometric functions of several variables by using algebraic de Rham cohomology and (co-)homology with local constant sheaf on the complement of algebraic hypersurfaces.

The book contains four chapters and four appendices. The first chapter is a short (19 pages) introduction of the Euler-Gauss hypergeometric function and its basic properties. The second chapter (80 pages) deals with “Representation of complex integrals and twisted de Rham cohomologies”. The third chapter (80 pages) deals with “Arrangement of hyperplanes and hypergeometric functions over Grassmannians” and the fourth chapter (77 pages) with “Holonomic difference equations and asymptotic expansion”. Each chapter is divided in numerous sections and subsections.

There are four appendices on “Mellin’s generalized hypergeometric functions”, “The Selberg integral and hypergeometric functions of BC type”, “Monodromy representation of hypergeometric functions of type \((2,m+1;\alpha)\)” and the already mentioned Appendix D.

The book ends with eight pages of references and three index pages. The text of the book is not an exhaustive textbook treatment of special functions. However, the authors construct – from the viewpoint that hypergeometric functions are complex integrals of complex powers of polynomials – a unified theory of hypergeometric functions of several variables by using algebraic de Rham cohomology and (co-)homology with local constant sheaf on the complement of algebraic hypersurfaces.

Reviewer: Roelof Koekoek (Delft)

##### MSC:

33-02 | Research exposition (monographs, survey articles) pertaining to special functions |

33C20 | Generalized hypergeometric series, \({}_pF_q\) |

14F40 | de Rham cohomology and algebraic geometry |