Theory of Hypergeometric Functions
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
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Kazuhiko Aomoto Michitake Kita
Theory of Hypergeometric Functions
Springer Monographs in Mathematics
For further volumes published in this series see www.springer.com/series/3733
Kazuhiko Aomoto
•
Michitake Kita
Theory of Hypergeometric Functions With an Appendix by Toshitake Kohno
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Kazuhiko Aomoto Professor Emeritus Nagoya University Japan [email protected]
Toshitake Kohno (Appendix D) Professor Graduate School of Mathematical Sciences The University of Tokyo Japan [email protected]
Michitake Kita (deceased 1995)
Kenji Iohara (Translator) Professor Universit´e Claude Bernard Lyon 1 Institut Camille Jordan France [email protected]
ISSN 1439-7382 ISBN 978-4-431-53912-4 e-ISBN 978-4-431-53938-4 DOI 10.1007/978-4-431-53938-4 Springer Tokyo Dordrecht Heidelberg London New York Library of Congress Control Number: 2011923079 Mathematics Subject Classification (2010): 14F40, 30D05, 32S22, 32W50, 33C65, 33C70, 35N10, 39B32 c Springer 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
One may say that the history of hypergeometric functions started practically with a paper by Gauss (cf. [Gau]). There, he presented most of the properties of hypergeometric functions that we see today, such as power series, a differential equation, contiguous relations, continued fractional expansion, special values and so on. The discovery of a hypergeometric function has since provided an intrinsic stimulation in the world of mathematics. It has also motivated the development of several domains such as complex functions, Riemann surfaces, differential equations, difference equations, arithmetic theory and so forth. The global structure of the Gauss hypergeometric function as a complex function, i.e., the properties of its monodromy and the analytic continuation, has been extensively studied by Riemann. His method is based on complex integrals. Moreover, when the parameters are rational numbers, its relation to the period integral of algebraic curves became clear, and a fascinating problem on the uniformization of a Riemann surface was proposed by Riemann and Schwarz. On the other hand, Kummer has contributed a lot to the research of arithmetic properties of hypergeometric functions. But there, the main object was the Gauss hypergeometric function of one variable. In contrast, for more general hypergeometric functions, including the case of several variables, the question ari
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