Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava

873 Pages·2014·4.57 MB·English

by  G. V.

Gradimir V. Milovanović
Michael Th. Rassias Editors
Analytic
Number Theory,
Approximation
Theory, and Special
Functions
In Honor of Hari M. Srivastava
Analytic Number Theory, Approximation Theory,
and Special Functions

Gradimir V. Milovanovic´ (cid:129) Michael Th. Rassias
Editors
Analytic Number Theory,
Approximation Theory,
and Special Functions
In Honor of Hari M. Srivastava
123
Editors
GradimirV.Milovanovic´ MichaelTh.Rassias
MathematicalInstitute ETH-Zürich
SerbianAcademyofSciencesandArts DepartmentofMathematics
Belgrade,Serbia Zürich,Switzerland
ISBN978-1-4939-0257-6 ISBN978-1-4939-0258-3(eBook)
DOI10.1007/978-1-4939-0258-3
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2014931380
MathematicsSubjectClassification(2010):05Axx,11xx,26xx,30xx,31Axx,33xx,34xx,39Bxx,41xx,
44Axx,65Dxx
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Preface
The volume “Analytic Number Theory, Approximation Theory, and Special
Functions” consists of 35 articles written by eminent scientists from the
international mathematical community, who present both research and survey
works. The volume is dedicated to professor Hari M. Srivastava in honor of his
outstandingworkinmathematics.
ProfessorHariMohanSrivastavawasbornonJuly5,1940,atKaroninDistrict
Ballia of the province of Uttar Pradesh in India. He studied at the University of
Allahabad, India, where he obtained his B.Sc. in 1957 and M.Sc. in 1959. He
receivedhisPh.D.fromtheJodhpurUniversity(nowJaiNarainVyasUniversity),
India,in1965.Hebegunhisuniversity-levelteachingcareerattheageof19.
H. M. Srivastava joined the Faculty of Mathematics and Statistics of the
University of Victoria in Canada as associate professorin 1969 and then as a full
professorin 1974. He is an EmeritusProfessor at the University of Victoria since
2006.
ProfessorSrivastavahasheldseveralvisitingpositionsinuniversitiesoftheUSA,
Canada,theUK,andmanyothercountries.
Hehaspublished21booksandmonographsandeditedvolumesbywell-known
international publishers, as well as over 1,000 scientific research journal articles
in pure and applied mathematicalanalysis He has served or currently is an active
member of the editorial board of several international journals in mathematics. It
is worth mentioning that he has published jointly with more than 385 scientists,
includingmathematicians,statisticians,physicists,andastrophysicists,fromseveral
partsoftheworld.ProfessorSrivastavahassupervisedseveralgraduatestudentsin
numerousuniversitiestowardstheirmaster’sandPh.D.degrees.
ProfessorSrivastavahasbeenbestowedseveralawardsincludingmostrecently
the NSERC 25-Year Award by the University of Victoria, Canada (2004), the
Nishiwaki Prize, Japan (2004), Doctor Honoris Causa from the Chung Yuan
Christian University, Chung-Li, Taiwan, Republic of China (2006), and Doctor
HonorisCausafromtheUniversityofAlbaIulia,Romania(2007).
ThenameofprofessorSrivastavahasbeenassociatedwithseveralmathematical
terms, including Carlitz–Srivastava polynomials, Srivastava–Panda multivariable
v
vi Preface
H-function, Srivastava–Agarwal basic (or q-) generating function, Srivastava–
Buschman polynomials, Chan–Chyan–Srivastava polynomials, Srivastava–Wright
operators, Choi–Srivastava methods in Analytic Number Theory, and Wu–
Srivastavainequalityforhighertranscendentalfunctions.
In this volume dedicated to professor Srivastava an attempt has been made to
discuss essential developmentsin mathematicalresearch in a varietyof problems,
mostofwhichhaveoccupiedtheinterestofresearchersforlongstretchesoftime.
Someofthecharacteristicfeaturesofthisvolumecanbesummarizedasfollows:
– Presentsmathematicalresultsandopenproblemsinasimpleandself-contained
manner.
– Containsnewresultsinrapidlyprogressingareasofresearch.
– Providesanoverviewofoldandnewresults,methods,andtheoriestowardsthe
solutionoflong-standingproblemsinawidescientificfield.
Thebookconsistsofthefollowingfiveparts:
1. Analytic Number Theory, Combinatorics, and Special Sequences of Numbers
andPolynomials
2. AnalyticInequalitiesandApplications
3. ApproximationofFunctionsandQuadratures
4. Orthogonality,Transformations,andApplications
5. SpecialandComplexFunctionsandApplications
Part I consists of nine contributions. A. Ivic´ presents a survey with a detailed
discussionofpowermomentsoftheRiemannzetafunction(cid:2).s/,whensliesonthe
“criticalline”<s D1=2.Itincludesearlyresults,themeansquareandmeanfourth
power,highermoments,conditionalresults,andsomeopenproblems.M. Hassani
givessomeexplicitupperandlowerboundsfor(cid:3) ,where0 <(cid:3) < (cid:3) <(cid:3) < (cid:2)(cid:2)(cid:2)
n 1 2 3
are consecutive ordinates of nontrivial zeros (cid:4) D ˇ C i(cid:3) of the Riemann zeta
function,includingthe asymptoticrelation(cid:3) log2n(cid:3)2(cid:5)nlogn (cid:4) 2(cid:5)nloglogn
n
as n ! 1. Y. Ihara and K. Matsumoto prove an unconditional basic result
related to the value-distributions of f.L0=L/.s;(cid:6)/g and of f.(cid:2)0=(cid:2)/.s C i(cid:7)/g ,
(cid:6) (cid:7)
where (cid:6) runs over Dirichlet characters with prime conductors and (cid:7) runs over
R. J. Choi presents a survey on recent developments and applications of the
simple and multiple gamma functions (cid:8) , including results on multiple Hurwitz
n
zeta functions and generalized Goldbach–Euler series. A. A. Bytsenko and E.
Elizaldeconsiderapartitionfunctionofhyperbolicthree-geometryandassociated
Hilbertschemes. In particular,the role of (Selberg-type)Ruelle spectral functions
of hyperbolic geometry for the calculation of partition functions and associated
q-series is discussed. Y. Simsek considers families of twisted Bernoulli numbers
and polynomials and their applications. Several relationships between Bernoulli
functions, Euler functions, some arithmetic sums, Dedekind sums, Hardy Berndt
sums, DC-sums, trigonometric sums, and Hurwitz zeta function are given. A. K.
AgarwalandM.Ranadiscussacombinatorialinterpretationofageneralizedbasic
series. Namely, using a bijection between the Bender–Knuth matrices and the n-
colorpartitionsestablishedbyAgarwal(ARSCombinatoria61,97–117,2001),they
Preface vii
extendarecentresulttoa3-wayinfinitefamilyofcombinatorialidentities.A.Sofo
proves some identities for reciprocal binomial coefficients, and M. Merca shows
thattheq-Stirlingnumberscanbeexpressedintermsoftheq-binomialcoefficients
andviceversa.
Part II is dedicated to analytic inequalities and several applications. Ibrahim
and Dragomir give a survey of some recent results for the celebrated Cauchy–
Bunyakovsky–Schwarz inequality for functions defined by power series with
nonnegative coefficients. G. D. Anderson, M. Vuorinen, and X. Zhang provide a
survey of recent results in special functions of classical analysis and geometric
function theory, in particular the circular and hyperbolic functions, the gamma
function,theellipticintegrals,theGaussianhypergeometricfunction,powerseries,
and mean values. M. Merkle gives a collection of some selected facts about the
completely monotone (CM) functions that can be found in books and papers
devotedto differentareasof mathematics. In particular,he emphasizesthe role of
representation of a CM function as the Laplace transform of a measure and also
presentsanddiscussesalittleknownconnectionwithlog-convexity.S.Abramovich
considers results on superquadracity, especially those related to Jensen, Jensen–
Steffensen, and Hardy’s inequalities. S. Ding and Y. Xing present an up-to-date
accountoftheadvancesmadeinthestudyofLp theoryofGreen’soperatorapplied
todifferentialforms,includingLp-estimates,LipschitzandBMOnorminequalities,
as well as inequalities with Lp.logL/˛ norms. B. Yang uses methods of weight
coefficients and techniques of real analysis to derive a multidimensional discrete
Hilbert-type inequality with a best possible constant factor. F. Qi, Q.-M. Luo,
andB.-N.Guoestablish sufficientandnecessaryconditionssuchthatthe function
.e˛t(cid:3)eˇt/=.e(cid:9)t(cid:3)e(cid:10)t/ismonotonic,logarithmicconvex,logarithmicconcave,3-log-
convex,and 3-log-concave on R. P. Cerone obtains approximationand boundsof
theGinimeandifferenceandprovidesareviewofrecentdevelopmentsinthearea.
Finally,M.A.Noorconsiderstheparametricnonconvexvariationalinequalitiesand
parametricnonconvexWiener–Hopfequations.Using the projectiontechnique,he
establishestheequivalencebetweenthem.
ApproximationoffunctionsandquadraturesistreatedinPartIII.N.K.Goviland
V. Gupta discussStancu-typegeneralizationof operatorsintroducedby Srivastava
and Gupta (Math. Comput. Modelling 37, 1307–1315, 2003). M. Mursaleen and
S. A. Mohiuddine prove the Korovkin-type approximation theorem for functions
of two variables, using the notion of statistical summability .C;1;1/, recently
introducedbyMoricz(J. Math.Anal.Appl.286,340–350,2003).In1961,Baker,
Gammel, and Wills formulated their famous conjecture that if a function f is
meromorphicin the unitballand analyticat 0,then a subsequenceof its diagonal
Padé approximantsconvergesuniformlyin compactsubsetsto f. This conjecture
wasdisprovedin2001,butitgeneratedanumberofrelatedunresolvedconjectures.
D.S. Lubinskyreviewstheirstatus. A.R. Hayotov,G. V.Milovanovic´,andK.M.
Shadimetov construct the optimal quadrature formulas in the sense of Sard, as
wellasinterpolationsplinesminimizingthesemi-norminthespaceK .P /,where
2 2
K2.P2/isaspacReoffunctions' which'0 isabsolutelycontinuousand'00belongs
toL .0;1/and 1.'00.x/C!2'.x//2dx < 1.Finally,asurveyonsomespecific
2 0
viii Preface
nonstandard methods for numerical integration of highly oscillating functions,
mainlybasedonsomecontourintegrationmethodsandapplicationsofsomekinds
of Gaussian quadratures, including complex oscillatory weights, is presented by
G.V.Milovanovic´andM.P.Stanic´.
In Part IV, C. Ferreira, J. L. López, and E. Pérez Sinusía study asymptotic
reductionsbetweentheWilsonpolynomialsandthelower-levelpolynomialsofthe
Askeyscheme,andK.Castillo, L.Garza,andF. Marcellánanalyzea perturbation
of a nontrivialprobability measure d(cid:10) supported on an infinite subset on the real
line,whichconsistsoftheadditionofatime-dependentmasspoint.A.F.Loureiro
andS.YakubovichconsiderspecialcasesofBoas–Buck-typepolynomialsequences
andanalyzesomeexamplesofgeneralizedhypergeometric-typepolynomials.P.W.
Karlsson gives an analysis of Goursat’s hypergeometric transformations, and A.
Kılıçmanconsiderspartialdifferentialequations(PDE)withconvolutiontermand
proposes a new method for solving PDE. Finally, using the fixed point method,
C. Park proves the Hyers–Ulam stability of the orthogonally additive–additive
functionalequation.
Specialfunctionsandcomplexfunctionswithseveralapplicationsarepresented
inPartV.Á.Baricz,P.L.Butzer,andT.K.Pogányhaveprovidedageneralization
of the complete Butzer–Flocke–Hauss (BFH) ˝-function in a natural way by
using two approaches and obtain several interesting properties. Á. Baricz and T.
K. Pogány consider properties of the product of modified Bessel functions and
establish discrete Chebyshev-type inequalities for sequences of modified Bessel
functions of the first and second kind. S. Porwal and D. Breaz investigate the
mapping properties of an integral operator involving Bessel functions of the first
kind on a subclass of analytic univalent functions. V. V. Mityushev introduces
and uses the Poincaré ˛-series (˛ 2 Rn) for classical Schottky groups in order
to solve Riemann–Hilbert problems for n-connected circular domains. He also
gives a fast algorithm for the computation of Poincaré series for disks that are
close to each other. N. E. Cho considers inclusion properties for certain classes
of meromorphic multivalent functions. He introduces several new subclasses of
meromorphicmultivalentfunctionsandinvestigatesvariousinclusionpropertiesof
thesesubclasses.Finally,I.LahiriandA.BanerjeediscusstheinfluenceofGross-
problemonthesetsharingofentireandmeromorphicfunctions.Itishopethatthe
bookwillbeparticularlyusefultoresearchersandgraduatestudentsinmathematics,
physics,andothercomputationalandappliedsciences.
Finally, we wish to express our deepest appreciation to all the mathematicians
from the international mathematical community, who contributed their papers for
publication in this volume dedicated to Hari M. Srivastava, as well as to the
refereesfortheircarefulreadingofthemanuscripts.Athankalsogoestoprofessor
MarijaStanic´ (UniversityofKragujevac,Serbia),forherhelpduringthetechnical
preparationofthe manuscript.Lastbutnotleast, we are verythankfulto Springer
foritsgeneroussupportforthepublicationofthisvolume.
Belgrade,Serbia GradimirV.Milovanovic´
Zürich,Switzerland MichaelTh.Rassias
Contents
PartI AnalyticNumberTheory,Combinatorics,andSpecial
SequencesofNumbersandPolynomials
TheMeanValuesoftheRiemannZeta-FunctionontheCriticalLine..... 3
AleksandarIvic´
ExplicitBoundsConcerningNon-trivialZerosoftheRiemann
ZetaFunction..................................................................... 69
MehdiHassani
On the Value-Distribution of Logarithmic Derivatives
ofDirichletL-Functions......................................................... 79
YasutakaIharaandKohjiMatsumoto
MultipleGammaFunctionsandTheirApplications......................... 93
JunesangChoi
On PartitionFunctions of Hyperbolic Three-Geometry
andAssociatedHilbertSchemes................................................ 131
A.A.BytsenkoandE.Elizalde
FamiliesofTwistedBernoulliNumbers, TwistedBernoulli
Polynomials,andTheirApplications .......................................... 149
YilmazSimsek
CombinatorialInterpretationofaGeneralizedBasicSeries................ 215
A.K.AgarwalandM.Rana
IdentitiesforReciprocalBinomials ............................................ 227
AnthonySofo
ANoteonq-StirlingNumbers.................................................. 239
MirceaMerca
ix