Table Of ContentVISTAS 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)
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VISTAS OF SPECIAL FUNCTIONS
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
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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
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To Professor Michel Waldschmidt with deep respect
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TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
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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 func-
tions. 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
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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
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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
