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VISTAS OF SPECIAL FUNCTIONS TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk VISTAS OF SPECIAL FUNCTIONS Shigeru Kanemitsu & Haruo Tsukada Kinki University, Japan World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I (cid:13)(cid:10) Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. VISTAS OF SPECIAL FUNCTIONS Copyright © 2007 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. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-270-774-1 ISBN-10 981-270-774-3 Printed in Singapore. ZhangJi - Vistas of Special.pmd 1 4/11/2007, 5:37 PM March27,2007 17:14 WSPC/BookTrimSizefor9inx6in vista To Professor Michel Waldschmidt with deep respect v March27,2007 17:14 WSPC/BookTrimSizefor9inx6in vista TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk March27,2007 17:14 WSPC/BookTrimSizefor9inx6in vista Preface This book is intended for aspirant readers who are eager to have basic knowledge of special functions in an organic way. We have kept paying at- tention to make an order in various equivalent statements on special functions. A unique feature is that the reader can gain a grasp of (almost) all existing(andscatteredaround)formulasinthe theoryof gammafunctions etc. in a clear perspective through the theory of zeta-functions. Thus, this is a book of special functions in terms of the zeta-functions. Reading through this book, the reader can master both (cid:12)elds e(cid:14)ciently. Here a hunter looking for two rabbits gets two. Here are some descriptions of the contents. InChapter1,wepresentauni(cid:12)edtheoryofBernoullipolynomialswith all equivalent conditions properly located. We have revealed that the dif- ference equation (DE) satis(cid:12)ed by the Bernoulli polynomial corresponds to di(cid:11)erentiation while the Kubert identity (K) corresponds to integration (the Riemann sum into equal division). This new view point makes the whole theory very lucid. InChapter2weshallpresentratherclassicalandstandardtheoryofthe gamma and related functions. Classical as it looks, we shall provide some very unique features of the Euler digamma function from which we may deducethecorrespondingpropertiesofthegammafunction. Especially,we shall give three proofs of the remarkable formula of Gauss on the values of the digamma function at rational arguments. One is classical and is presented in Chapter2. Othertwoproofsaremoreoriginalgivenin Chap- ter 8, one is the limiting case (Theorem 8.2) of the Eisenstein formula in itsgenuineform(atheoremduetoH.-L.Li,L.-P.DingandM.Hashimoto, describingabasiselementintermsofanotherbasisofthespaceofperiodic Dirichlet series), the other is the theorem of M. Hashimoto, S. Kanemitsu vii March27,2007 17:14 WSPC/BookTrimSizefor9inx6in vista viii Vistasof Special Functions and M. Toda about the equivalence between the (cid:12)nite form of the value of the Dirichlet L-function at 1 and the formula of Gauss. In Chapter 3, we shall present the theory of the Hurwitz zeta-function. The mainingredientis the integralrepresentationforits partialsum. This is to the e(cid:11)ect that once we have an integral representation as the one we have, we may immediately draw information for the derivatives, i.e. we have an inheritance of the information. The integral representation for the partial sum is so informative that it contains all information we need (Theorem3.1). TheversatilityofthisresultwillbedevelopedinChapter5, wherethroughLerch’sformula,wetransfertheresultsontheHurwitzzeta- function to those on the gamma and related functions. Especially, the asymptoticresultsestablishedinChapter3willimmediatelytransfertothe Stirling formula and other asymptotic formulas for relatives of the gamma function. In Chapter 4, we shall present the theory of Bernoulli polynomials through the negative integer values (cid:16)( n;z) of the Hurwitz zeta-function. (cid:0) Here we shall establish only three statements, i.e. the Fourier series (H), thedi(cid:11)erenceequation(DE)andtheKubertidentity(K)fromanyofwhich we may complete the theory following the logical scheme in Chapter 1. In Chapter 5, (cid:12)rst we shall reveal the power of theorems in Chapter 3 toexhibitwhattheDufresnoy-Pisottypeuniquenesstheoremmeans. Then we shall go on to presenting the (cid:12)rst circle (krug p’iervyi) which connects variousidentities betweengammaand trigonometricfunctions to the func- tionalequations(zeta-symmetry)ofthezeta-functions. Thusweshallshow that everything comes from the functional equation. A remarkable notice is that such trigonometric identities like the in(cid:12)nite product for the sine function or the partial fraction expansion for the cotangent function are equivalent to the functional equation, thus revealing why Euler succeeded in solving the Basler problem. In Chapter 6, we shall further pursue this zeta-symmetryin relation to the crystal symmetry through the Epstein zeta-function. We surpass the preceding results by introducing the signs and giving the Chowla-Selberg type formula (based on the Mellin-Barnes integrals) and provide a quick means for computation of the Madelung constants. In Chapter7, weshallproviderudiments of the theoryof Fourierseries andintegralstosuchanextentthatissu(cid:14)cientforapplicationsandreading throughthis book, for the sakeof the readerwho wantsto learnit quickly. Chapter 8 is, so to say, a discrete version of Chapter 7, i.e. the (cid:12)nite Fourier series (transforms). Through this we make clear the orthogonality March27,2007 17:14 WSPC/BookTrimSizefor9inx6in vista Preface ix of characters and other bases of the space of Dirichlet series with periodic coe(cid:14)cients, giving rise to the theorem mentioned above. We can naturally extend our method to develop the similar theory for higher derivatives of the Dirichlet L-function, including Kronecker’slimit formula. But because of limitation of time, we cannot go further. Appendix A gives the very basics of the theory of complex functions. We present mostly results only, and the interested reader should consult a standardbookfortheirproofs. Weshallgive,however,somedetailsonthe use of residue theorem. Appendix B assembles summation formulas and convergence theorems used in the book. Especially, the Fourier series for the (cid:12)rst periodic Bernoulli polynomial is so essential and important, we give two proofs, one depending on ordinary Fourier theory (Chapter 7) and the other on the polylogarithm function of degree 1, where we apply the theorem of Abel and Dirichlet in place of Fourier theory. As is explained above, Chapters 1 and 4 are parallel, so are Chapters2 and 5. To understand Chapters 4 and 5, one should read Chapter 3 (cid:12)rst. If one (cid:12)nds some di(cid:14)culties, then one is referred to Appindices A and B. Chapters7and8canbereadindependently,butitwillbemoreinstructive to read both in parallel. Chapter 6 can be read separately which requires more knowledge of Bessel functions. Because of lack of time, we could not state much about them. This publication wassupported by Kinki University Grantfor Publica- tion, No. GK04 in the academic year 2006. The authors are thankful to Kinki University for their generosity of this support. They also would like tothankMs. ChiewYingOiwhohelpedthemallthroughtheprocesswith her e(cid:14)cient editorial skills. And toward the end of the process Ms. Zhang Ji supported us and we would like to express our heartily thanks to her. The authors would like to express their hearty thanks to their close friendProfessorY.Tanigawaforhisconstantsupport,encouragement,and stimulatingdiscussions. The(cid:12)rstauthorwouldliketothankhisclosefriend Professor Heng Huat Chan for his enlightening remark on the equivalent statements to the functional equation, thanks to which he got motivated enough to start writing this book. The second author was naturally got infectedthepassionofthe(cid:12)rst. ThanksarealsoduetoMs. L.-P.Dingand Mr. M. Toda for their devoted endeavor, without their enthusiastic help, the book would have not been risen out. the authors