The double Mellin-Barnes type integrals and their applications to convolution theory PDF
Preview The double Mellin-Barnes type integrals and their applications to convolution theory
Series on Soviet and East European Mathematics Vol. 6 The Double Mellin-Barnes Type Integrals and their Applications to Convolution Theory This page is intentionally left blank Series on Soviet and East European Mathematics Vol. 6 The Double Mellin-Barnes Type Integrals and their Applications to Convolution Theory Nguyen Thanh Hai & S B Yakubovich Byelorussian State University World Scientific Singapore • New Jersey • London • Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH Series on Soviet and East European Mathematics THE DOUBLE MELLION-BARNES TYPE INTEGRALS AND THEIR APPLICATIONS TO CONVOLUTION THEORY Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN 981-02-0690-9 Printed in Singapore by JBW Printers & Binders Pte. Ltd. PREFACE This book presents new results on the theory of double Mel1in- Barnes type integrals and their applications to convolution theory. This class of integrals is known as the H-function of two variables and in the most general case it was first introduced by R.G.Buschman in 1978. In an attempt to make the book self-contained, paragraph §1 of Chapter I provides the necessary brief historical background material in the theory of simple and double Mellin-Barnes integrals. In Chapter I we give the definition and the main properties of the H-function of two variables in the general case. In paragraphs §3 — §4 we first present the complete solution of the convergence problem of the general H- function of two variables. In the following paragraphs (§5—§8) we discuss various fundamental properties of the general H-function: contiguous relations, the double Mellin transform, series representa tions. In the last paragraph §9 of Chapter I in order to classify the H-functions of two variables we introduce the notion of characteristic which will be used in the following Chapters. In Chapter II we introduce and study the H-function of two varia bles with the third characteristic and its special case — the G-function of two variables. These functions are particular cases of the general H-function and they have immediate applications for studying the convolution theory later on. Here, besides the convergence theorems we give various properties, which are habitual only for these functions. The list of special cases of the G-function of two variables is obtained in §13. In Chapter III we present the modern method to study the H- and G-integral transforms together with their generalizations. Here we consider these transforms in the special space 9K~ (L) which is very v convenient to obtain the inversion theorems and it allows us to describe the composition structure of the mentioned transforms. Various particu lar cases of the G-transform are given. In Chapter IV we construct and study the general integral convolu tions involving the classical Laplace convolution as special case. It gives rather a simple method to obtain the integral convolutions for Mellin type transforms. Many examples of convolutions for various known transforms are given. Here are considered new applications of known convolutions to evaluation of series and integrals. For the sake of convenience, we give author, subject and notation indices in the end of the book. This book is written primarily for teachers, researchers and graduate students in the areas of special functions and integral transforms. In this book research workers and users in the field of special functions of two variables will find new fundamental information and its application to the convolution theory. Many persons have made a significant contribution to this book, both directly and indirectly. Contribution of subject matter is duly acknowledged throughout the text and in the bibliography. We are especially thankful to Professors Robert G. Buschman of the University of Wyoming, USA, Hari M. Srivastava of the University of Victoria, Canada, and Megumi Saigo of the University of Fukuoka, Japan, for their keen support throughout the subject of this book and for sending us relevant reprints and preprints of their works. This book is written during the academic year 1990—1991 at the "Research Scientific Laboratory of Applied Methods of Mathematical Analysis" of the Byelorussian State University, where both authors work. We are immensely indebted to Professor Oleg I.Marichev, who was bur scientific supervisor, for his constant encouragement during the last decade, when we studied at the Byelorussian State University. vi Finally, we are pleased to thank Mrs. Dr. Lyudmila K.Bizyuk for reading the manuscript and for suggesting a number of invaluable improvements. June 1991 Dr. Nguyen Thanh Hai Byelorussian State University Dr. Semen B.Yakubovich Minsk-80, USSR vii This page is intentionally left blank CONTENTS Chapter I. General H-function of Two Variables § 1. Historical background 1 § 2. Definition and notations 9 § 3. The convergence region of the general H-function of two variables 12 § 4. The H-function of two real positive variables 23 § 5. Simple contiguous relations for the H-function of two variables 47 § 6. Main properties for the H-function 51 § 7. The double Mel 1 in transform 54 § 8. Series representations for the H-function of two variables 57 § 9. Characteristic of the general H-function of two variables 69 Chapter II. The H-function of two variables with the third characteristic § 10. Definition and notations 72 § 11. Convergence theorems 75 § 12. Reduction formulas for the H-function with the third characteristic 80 § 13. The G-function of two variables and its special cases...89 § 14. The double Kampe de Feriet hypergeometric series 103 ix Chapter III. One-dimensional H-transform and its composition structure § 15. Spaces 3H"1 (L) and JJT1 (L) 119 § 16. One-dimensional H-transform in the spaces 3JT1 (L) and UK"1 (L) 129 § 17. The G-transform and its special cases 142 § 18. Composition structure of the H- and G- transforms 153 Chapter IV. General integral convolutions for the H-transform § 19. Classical Laplace convolution and its new properties...162 § 20. General integral convolution: definition, existence and factorization property 170 § 21. Typical examples of the general convolutions 181 § 22. Case of the same kernels: the general Laplace convolution 190 § 23. G-convolution and its typical examples 198 § 24. Convolutions for some classical integral transforms.... 209 § 25. Modified H-convolution 225 § 26. General Leibniz rules and their integral analogs 233 Bibliography 261 Author Index 279 Subject Index 285 Notations 291 x
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