Series of Bessel and Kummer-Type Functions

212 Pages·2017·2.156 MB·english

by  Árpád Baricz, Dragana Jankov Maširevic, Tibor K. Pogány

Árpád Baricz (cid:129) Dragana Jankov Maširevic´ (cid:129)
Tibor K. Pogány
Series of Bessel and
Kummer-Type Functions
123
ÁrpádBaricz DraganaJankovMaširevic´
JohnvonNeumannFacultyofInformatics DepartmentofMathematics
InstituteofAppliedMathematics JosipJurajStrossmayerUniversityofOsijek
ÓbudaUniversity Osijek,Croatia
Budapest,Hungary
DepartmentofEconomics
Babes¸–BolyaiUniversity
Cluj–Napoca,Romania
TiborK.Pogány
FacultyofMaritimeStudies
UniversityofRijeka
Rijeka,Croatia
JohnvonNeumannFacultyofInformatics
InstituteofAppliedMathematics
ÓbudaUniversity
Budapest,Hungary
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-74349-3 ISBN978-3-319-74350-9 (eBook)
https://doi.org/10.1007/978-3-319-74350-9
LibraryofCongressControlNumber:2018932413
MathematicsSubjectClassification(2010):40A30,40C10,33C10,33C15,41A30
©SpringerInternationalPublishingAG,partofSpringerNature2017
Preface
The summation of series of special functions (or, accepting Turán’s intervention,
“usefulfunctions”)isasubdisciplineofClassicalAnalysis.Functionalseriesbuilt
frommembersofthe,so-called,Besselfunctionfamilyplayaparticularlyimportant
role in this field. The Bessel function family includes a vast range of functions:
Bessel functions of the first and second kind, modified Bessel functions of the
firstandsecondkind,Hankelfunctions,StruveandmodifiedStruvefunctions,von
Lommelfunctions,forinstance.Thereisalsoanextensiveliterature,includingthe
monumental monograph [333], concerned with important properties and the vast
rangeofapplicationsofsuchfunctionsandvariousfunctionalseriesbuiltfromthem.
AnimportanttopicwithinthetheoryofBesselfunctionsisthestudyoffunctional
seriesofBesselandrelatedfunctions,whoseroleinmathematicalphysics,science,
astronomy,andengineeringisimmense.
TheclassesofinfiniteseriesexploredinthismonographareNeumann,Kapteyn,
Schlömilch, and Dini series, whose terms contain certain members of the Bessel
function family or special functions that arise from the class of hypergeometric
functions(Kummerfunction).Thebuildingblocksoftheseseriesdependoncertain
parameters. So, in short, the main difference between these series is that in terms
oftheNeumannseriesthesummationindexistheorder(parameter)oftheBessel
function;intermsoftheKapteynseriesthesummationindicesaretheorderandthe
argument,while in terms of the Schlömilch series the argumentis the summation
index. Also, using similar criteria, one can define general Neumann, Schlömilch,
and Kapteyn series of hypergeometric or other special functions, guided by the
above classification principle. On the other hand, the coefficient of the argument
inaDiniseriesinvolvesthezerosoftheinitialfunctionfromtheBesselfamily,or
thoseoftherelatedDinifunction.
Functions in the Bessel family and the Kummer function have either power
seriesordefiniteintegralrepresentationsortheyareparticularsolutionsofordinary
differentialequations. Thuswe shall adopta three-prongedapproachin our study
andwillexploresummationsofsums,summationsofintegrals,andsummationsof
functions that are solutions of Bessel, Struve, Kummer, or certain other classical
ordinarydifferentialequations.Whileweareaddressingmainlythesameproblems
assomeofthegreatforefathersofthefieldofFourier–Besselseries,includingCarl
GottfriedNeumann(1832,Königsberg–1925,Leipzig);WillemKapteyn(Kapteijn)
(1849,Barneveld–1927,Utrecht); Oscar Xavier Schlömilch (1823,Weimar–1901,
Dresden);andinparallelUlisseDini(1845,Pisa–1918,Pisa),ourapproachtothese
considerationsissignificantlydifferent.
Baricz and Pogány in [20, p. 815, Theorem 3.2] introduced a method, which
completely reorganizes the classification “Fourier–Bessel series of the first type”
(where one input Bessel family member function occurs in terms of the series)
versus “Fourier–Bessel series of the second type” (where products of two or
more Bessel-like functions appear in terms of the series). More precisely, Bar-
icz and Pogány have incorporated all input functions in the products except
a chosen one, which is included into the coefficient, and they consider the
initial Fourier–Bessel series as the “series of the first type” with the newly
constituted coefficients. The importance of these results is further seen by bear-
ing in mind various new findings concerning derivatives of the Bessel func-
tion family with respect to the order posted on the Wolfram Functions web-
site(http://blog.wolfram.com/2016/05/16/new-derivatives-of-the-bessel-functions-
have-been-discovered-with-the-help-of-the-wolfram-language/).
We appreciatethatthe title ofa monographshouldbe conciseandinformative,
andnot“toolong.”Tocoverthephrase“Neumann,Kapteyn,SchlömilchandDini
SeriesofBesselFunctionsorHypergeometricTypeFunctions,”whichisaprecise
but excessively long title for a book, we adopted “Fourier–Bessel Series” as a
working title, influenced by the title of the article [145], and, e.g., by the title of
section XVIII, “Series of Fourier–Bessel and Dini” in the monograph [333] by
Watson. His presentation significantly differs from ours; we will briefly present
thistreatmentoffunctionsbyFourier–Besselseries, whichactuallybelongstothe
class of Schlömilch series, in the related subsection of the introductory chapter,
emphasizingthatwetreatFourier–BesselseriesinaweakersensethanKapteynand
Watson. We also note the fact that Bessel functions are linked to hypergeometric
functions;see,also,[314, Chapter8].So,thetitle “SeriesofBesselandKummer-
Type Functions” interpolates the previously mentioned two descriptions of the
contentsofthismonograph.
The starting point for our research was the study [249] by Pogány and Süli
in 2009 on Neumann series of Bessel functions of the first kind J(cid:2) and von
LommelfunctionsinwhichanintegralexpressionwasderivedforNeumannseries.
There, the cornerstones of the study were Dirichlet series associated with the
inputFourier–BesselseriesandtheLaplaceintegralofthisDirichletseries. While
proceeding with our research on mathematical tools associated with appropriate
Bessel-typehomogeneousandnonhomogeneousordinarydifferentialequations,we
extendedourstudy,whichthenresulted,amongothers,inthePh.D.thesisofJankov
Maširevic´ [130]in2011andthehabilitationthesis[244]ofPogányin2015.Those
twothesesarosefromseveraljointorseparatepublicationsandconstitutea major
partofthismonograph.
Our main objective in this monograph is to give a systematic overview of our
results concerning such series; textual material is gathered from diverse sources
including journal articles, theses, and conference papers, which had not appeared
beforeintheformofabook.
The bookis aimed at a mathematicalaudience,graduatestudents, and those in
thescientificcommunitywithinterestinanewperspectiveonFourier–Besselseries,
andtheirmanifoldandpolyvalentapplications,mainlyingeneralclassicalanalysis,
appliedmathematics,ormathematicalphysics.
A general introductionto the subject will be found in Chap.1, together with a
necessarily short overviewof special functions,Dirichlet series, Cahen’s formula,
and the Euler–Maclaurinsummation formula, among others, as it is assumed that
readershaveageneralbackgroundinrealandcomplexanalysis,andpossesssome
familiarity with functional analysis. Then, results on Neumann–Bessel series are
collected in the identically entitled Chap.2, followed by Chap.3, Kapteyn series,
where, in addition to Kapteyn–Bessel series, also Kapteyn–Kummer series are
presented. Chapter 4 focuses on Schlömilch–Bessel series and Schlömilch series
ofthep-extendedMathieuseries,whichrepresentsatransitiontoChap.5,entitled
Miscellanea, where Dini–Bessel series, Neumann and Kapteyn series of Struve
and modified Struve functions, and Neumann series of Jacobi polynomials are
considered. The main body of the book ends with a short overview of Neumann
seriesofMeijerG functions,whichisfollowedbyanexhaustivelistofreferences
andanIndex.Wenotethatadetailedoverviewofdiverseapplications,withlinksto
furtherrelevantsources,isgivenintheintroductorypartofeachchapter.
Besides the pure mathematical aspects of the obtained results, many potential
applicationitemsexist,e.g.,theKapteynseries’applicationsinvariousproblemsof
mathematicalphysics,e.g.,Kepler’sequation,pulsarphysics,andelectromagnetic
radiation;Neumannseries’useininfinitedielectricwedgeproblem,descriptionof
internal gravity waves in a Boussinesq fluid, propagation properties of diffracted
light beams, the orbital angular momentum quantum number, the wave functions
thatdescribethestatesofmotionofchargedparticlesinaCoulombfield,inversion
probabilityofalargespin,evaluationofthecapacitancematrixofasystemoffinite-
lengthconductors,modelingofthefreevibrationsofawoodenpole,andanalysisof
anisotropicmediumcontainingacylindricalboreholeareroutineprocedures.These
numericalcalculationsmainlytakeintoaccounttruncationofinfiniteseries.Instead,
thederivedintegralexpressionsmayleadto numericalquadratureimplementation
forwhichnumerousin-builtsoftwareroutinesarewidelydeveloped.
The authors take great pleasure in thanking Endre Süli (Oxford) for taking
part in the research endeavor, which initiated and now finally encompasses this
manuscript.WearealsoverygratefultoPaulLeoButzer(Aachen),DiegoDominici
(New Paltz), SaminathanPonnusamy(Chennai),and SanjeevSingh (Chennai)for
numerousvaluablesuggestions,remarks,anddiscussions,whichresultedincrucial
improvementsoftheexposition.
Székelyudvarhely ÁrpádBaricz
Trpinja DraganaJankovMaširevic´
Sušak TiborK.Pogány
October2017
Survey
Theaimofthisbriefsurveyistopresentashortoverviewofthetopicsdiscussedin
thisbook.
BesselFunctions Besselfunctionsaresolutionstothesecond-orderlinearhomo-
geneous Bessel differential equation. Discovered by the mathematician Daniel
Bernoulli and studied systematically by the astronomer Friedrich Bessel, Bessel
functionsappearfrequentlyinproblemsofappliedmathematics.Theyareparticu-
larlyimportantinproblemsassociatedwithwavepropagationandstaticpotentials.
Besselfunctionsofintegerorderarealsoknownascylinderfunctionsorcylindrical
harmonics, because they arise in the solution of Laplace’s equation in cylindrical
coordinates.AlthoughthestudyofBesselfunctionsispartofclassicalanalysis,their
beautifulpropertiesarecontinuallyexploredbynumerousresearchers,andseveral
newpropertiesarereportedeachyear.G.N.Watson’sbookAtreatiseonthetheory
of Bessel functions [333], written almost one hundred years ago, is an important
book in the theory of special functions, especially on topics associated with
asymptoticexpansions,series,zeros,andintegralsofBesselfunctions.Nowadays,
Watson’s book is a classic, and because of their remarkable properties, special
functions,such as Bessel functions,are frequentlyused also in probabilitytheory,
statistics,mathematicalphysics,andintheengineeringsciences.See,forexample,
theinterestingbookbyB.G.Korenev Besselfunctionsandtheirapplications,[156].
SeriesofBesselFunctions InfiniteseriesinvolvingdifferentkindsofBesselfunc-
tions occur quite frequentlyin both mathematicaland physicalanalysis. Watson’s
treatisecontainsfourchaptersondifferentkindsofseriesofBesselfunctions,such
asNeumann,Kapteyn,Fourier–Bessel,Dini,andSchlömilchseries.Becauseofthe
rangeofapplicationsinconcreteproblemsofappliedmathematics,seriesofBessel
functionshavebeenconsideredfrequentlybyresearchers.
The Topics Discussed in This Book In this book our aim is to establish certain
integralrepresentationsforNeumann,Kapteyn,Schlömilch,Dini,andFourierseries
of Bessel and other special functions, such as Struve and von Lommel functions.
Our objective is also to find the coefficients of the Neumann and Kapteyn series,
as well as closed-form expressions, and summation formulae for the series of
Besselfunctionsconsidered.Inthestudytheso-calledEuler–Maclaurinsummation
formula(whichisabeautifulbridgebetweencontinuousanddiscrete),theLaplace–
Stieltjes integral representation of Dirichlet series, and various bounds for Bessel
and Bessel-type functions (Struve, modified Struve, modified Bessel functions of
thefirstandsecondkind,vonLommelfunctions,andBesselfunctionofthesecond
kind) play an important role. Some integral representations are also deduced by
usingtechniquesfromthetheoryofdifferentialequations.
Contents
1 IntroductionandPreliminaries............................................. 1
1.1 TheGammaFunction ................................................. 3
1.1.1 Psi(orDigamma)Function.................................. 4
1.1.2 TheBetaFunction ........................................... 5
1.1.3 ThePochhammerSymbol................................... 6
1.2 BernoulliPolynomialsandNumbers................................. 6
1.3 Euler-MaclaurinSummationFormula................................ 7
1.4 DirichletSeriesandCahen’sFormula................................ 8
1.5 Mathieu.a;(cid:2)/-Series.................................................. 9
1.6 BesselDifferentialEquation .......................................... 10
1.7 BoundsUponJ(cid:2).x/.................................................... 11
1.8 BesselFunctionsFamily .............................................. 13
1.9 StruveDifferentialEquation .......................................... 16
1.10 SeriesBuiltbyBesselFunctionsFamilyMembers.................. 17
1.11 Fourier–BesselandDiniSeries ....................................... 18
1.12 HypergeometricandGeneralizedHypergeometricFunctions....... 19
1.12.1 GaussianHypergeometricFunction......................... 19
1.12.2 GeneralizedHypergeometricFunction ..................... 20
1.12.3 Fox–WrightGeneralizedHypergeometricFunction ....... 21
1.13 FurtherHypergeometricTypeFunctions............................. 22
1.14 Hurwitz–LerchZetaFunction......................................... 23
1.15 FractionalDifferintegral............................................... 23
2 NeumannSeries.............................................................. 27
2.1 IntegralRepresentationforNeumannSeriesofBessel
Functions............................................................... 29
2.1.1 BivariatevonLommelFunctionsasNeumannSeries...... 31
2.2 OnCoefficientsofNeumann–BesselSeries.......................... 32
2.2.1 Examples..................................................... 36
2.3 IntegralRepresentationsforN(cid:2).x/viaBessel
DifferentialEquation.................................................. 37
2.3.1 TheApproachbyChessin ................................... 37
2.3.2 SolvingBesselDifferentialEquationbyFractional
Integration.................................................... 41
2.3.3 FractionalIntegralRepresentation .......................... 43
2.4 IntegralRepresentationsforNeumann–BesselTypeSeries......... 45
2.4.1 IntegralFormoftheFirstTypeNeumannSeriesM(cid:2).x/... 46
2.4.2 IntegralFormofSecondTypeNeumannSeries
J(cid:2).x/;X(cid:2).x/ .................................................. 49
2.5 IntegralFormofNeumannSeriesNa;b.x/ ........................... 56
(cid:3);(cid:2)
2.5.1 SecondTypeNeumannSeriesN˛(cid:3)I;(cid:4)(cid:2).x/andN˛(cid:3)K;(cid:2)(cid:2).x/....... 60
2.6 PropertiesofProductofModifiedBesselFunctions................. 61
2.6.1 DiscreteChebyshevInequalities ............................ 62
2.6.2 IntegralFormofRelatedSecondTypeNeumannSeries... 64
2.6.3 IndefiniteIntegralExpressionsforSecondType
NeumannSeriesN(cid:2)(cid:3);(cid:2).x/..................................... 67
2.7 Summation Formulae for the First and Second Type
NeumannSeries........................................................ 69
2.7.1 ClosedFormoftheFirstTypeNeumannSeriesNI;˙..... 71
(cid:3);(cid:2)
2.7.2 Confluent Hypergeometric Functions
andSrivastava–DaoustFunction ............................ 76
2.8 NeumannSeriesRegardingthe(cid:5)02.a/Distribution.................. 78
n
2.9 ConnectingFirstandSecondTypeNeumannSeries................. 83
3 KapteynSeries ............................................................... 87
3.1 OnConvergenceofGeneralizedKapteynExpansion................ 90
3.2 IntegralRepresentationofKapteynSeries ........................... 93
3.3 AnotherIntegralFormofKapteynSeriesThroughBessel
DifferentialEquation.................................................. 95
3.4 IntegralExpressionofSpecialKindKapteynSeries................. 98
3.5 OnCoefficientsofKapteynSeries.................................... 100
3.5.1 Examples..................................................... 103
3.6 OnKapteyn–KummerSeries’IntegralForm......................... 104
3.6.1 TheMasterIntegralRepresentationFormula............... 106
3.6.2 TheNeumann–KummerandSchlömilch–Kummer
Series ......................................................... 110
4 SchlömilchSeries ............................................................ 113
4.1 IntegralRepresentationofSchlömilchSeries........................ 117
4.2 AnotherIntegralRepresentationofSchlömilchSeries .............. 118
4.3 SchlömilchSeriesBuiltbyModifiedBesselK(cid:2) ..................... 120
4.3.1 ClosedExpressionsforS2k(cid:2)2.0;x;(cid:2)/andS0.a;x;(cid:2)/..... 121
4.4 ConnectionBetweens(cid:2)2.0;x;(cid:2)/andGeneralized
MathieuSeries......................................................... 126
4.5 p-ExtendedMathieuSeriesasSchlömilchSeries.................... 130
4.5.1 ConnectionBetweenS(cid:3);p.r/andSchlömilchSeries
ofJ(cid:2) (cid:2)K(cid:3) ..................................................... 131
4.5.2 S(cid:3);p.r/andtheSchlömilchSeriesofK(cid:2)Terms............. 134
4.6 IntegralFormofPopov’sFormula(4.7).............................. 136
5 Miscellanea ................................................................... 139
5.1 TheFourier–BesselSeriesAssociatedwithStruveFunctions....... 139
5.2 SummationsofSeriesBuiltbyModifiedStruveFunction .......... 140
5.2.1 L(cid:2) asaNeumannSeriesofModified
BesselIFunctions ........................................... 143
5.3 Integralsof˝.x/-FunctionandMathieuSeriesviaTI;L.x/ ........ 149
(cid:2)
5.4 DifferentialEquationsforKapteynandSchlömilch
SeriesofI(cid:2);L(cid:2) ......................................................... 152
5.5 Bromwich–WagnerIntegralFormofJ(cid:2).x/........................... 159
5.6 SummingupSchlömilchSeriesofStruveFunctions................ 162
5.7 DiniSeries ............................................................. 165
5.8 DiniSeriesandtheBesselDifferentialEquation .................... 168
5.9 JacobiPolynomialsinSum ........................................... 169
5.10 SchlömilchSeriesofvonLommelFunctions........................ 174
5.10.1 ClosedFormExpressionsforS˛ .x/ ...................... 176
(cid:3);(cid:2)
5.11 NeumannSeriesofMeijerGFunction............................... 180
References......................................................................... 185
Index............................................................................... 199