Special Functions for Applied Scientists A.M. Mathai • Hans J. Haubold Special Functions for Applied Scientists 123 A.M.Mathai HansJ.Haubold McGillUniversity OfficeforOuterSpaceAffairs Montreal,QC UN,Vienna Canada Austria and [email protected] CentreforMathematicalSciences PalaCampus,Kerla India [email protected] ISBN:978-0-387-75893-0 e-ISBN:978-0-387-75894-7 DOI:10.1007/978-0-387-75894-7 LibraryofCongressControlNumber:2007942158 (cid:1)c 2008SpringerScience+BusinessMedia,LLC Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper. 9 8 7 6 5 4 3 2 1 springer.com Preface ThefirstS.E.R.C.(ScienceandEngineeringResearchCounciloftheDepartmentof ScienceandTechnology,GovernmentofIndia,NewDelhi)SchoolonSpecialFunc- tionswassponsoredbyDST(DepartmentofScienceandTechnology),NewDelhi, andconductedbyCMS(CentreforMathematicalSciences)forsixweeksin1995. The second S.E.R.C. School on Special Functions and Functions of Matrix Argu- mentwassponsoredbyDST,Delhi,andconductedbyCMSforfiveweeksduring theperiod29thMayto30thJune2000.InthesecondSchool,themainlectureswere givenbyDr.H.L.ManochaofDelhi,India,Dr.S.BhargavaofMysore,India,Dr. K.SrinivasaRaoandDr.R.JagannathanofChennai,IndiaandDr.A.M.Mathaiof Montreal, Canada. Supplementary lectures were given and problem sessions were supervised by Dr. K.S.S. Nambooripad and Dr. S.R. Chettiyar of CMS, Dr. R.N. Pillai(retired),Dr.T.S.K.MoothathuandDr.YageenThomasoftheUniversityof Kerala and Dr. E. Krishnan of the University College Trivandrum. Lecture Notes were brought out as Publication No.31 of CMS soon after the School was com- pleted.ThefirsttwoSchoolswereconductedinTrivandrum(Thiruvananthapuram) area. Dr. A.M. Mathai was the Director and Dr. K.S.S. Nambooripad was the Co- DirectorofthesetwoSchools. ThethirdS.E.R.C.SchoolonSpecialFunctionsandFunctionsofMatrixArgu- ment:RecentDevelopmentsandRecentApplicationsinStatisticsandAstrophysics, sponsored by DST, Delhi, was conducted for five weeks from 14th March to 15th April 2005 by CMS at its Pala Campus, Kerala, India. This time DST, wanted the lecturenotestobecollectedfromthemainlecturersinadvance,compiledanddis- tributedpriortothestartoftheSchool.Themainlecturesforthe3rdSchoolwere given by Dr. Hans J. Haubold of the Office of Outer Space Affairs of the United Nations,Dr.SergeB.Provost,ProfessorofActuarialSciencesandStatisticsofthe University of Western Ontario, Canada, Dr. R.K. Saxena of Jodhpur, India, Dr. S. BhargavaofMysore,IndiaandDr.A.M.MathaiofMontreal,Canada.Supplemen- tary lectures were given by Dr. A. Sukumaran Nair (Chairman, CMS), Dr. K.K. Jose(Director-in-Charge,CMSPalaCampus),Dr.R.N.Pillai,Dr.YageenThomas, Dr. V. Seetha Lekshmi, Dr. Alice Thomas, Dr. E. Sandhya, Dr. S. Satheesh and v vi Preface Dr.K.Jayakumar.ProblemsessionsweresupervisedbyDr.SebastianGeorgeand otherlecturers.Extratrainingintheuseofstatisticalpackages,LATEXandMath- ematica/MapleweregivenbyJoyJacob,SeemonThomasandK.M.Kurianofthe DepartmentofStatisticsofSt.ThomasCollege,Pala.Aspeciallecturesequenceon MatlabwasconductedbyAlexanderHauboldofColumbiaUniversity,U.S.A. The 4th S.E.R.C. School on Special Functions and Functions of Matrix Argu- ment:RecentAdvancesandApplicationstoStochasticProcesses,StatisticsandAs- trophysics,sponsoredbyDST,Delhi,wasconductedbyCMSforfiveweeksfrom 6thMarchto7thApril2006atitsPalaCampus.Themainlectureswerescheduled tobegivenbyDr.HansJoachimHauboldoftheOfficeofOuterSpaceAffairsofthe UnitedNations,Dr.P.N.RathieoftheUniversityofBrasilia,Brazil,Dr.S.Bhargava ofMysore,India,Dr.R.K.SaxenaofJodhpur,India,Dr.K.K.JoseofPala,Indiaand Dr.A.M.MathaiofMontreal,Canada.SupplementarylecturesweregivenbyDr.A. Sukumaran Nair (Chairman, CMS), Dr. K.S.S. Nambooripad (Director-in-Charge, CMSTrivandrumCampus),Dr.R.YDenisofGorakhpur,India,Dr.N.Unnikrish- nanNair(formerVice-ChancellorofCochinUniversityofScienceandTechnology, Kerala, India), Dr. Yageen Thomas and Dr. K. Jayakumar. One week was devoted toStochasticProcessesandrecentadvancesinthisareainthe4thSchool.Problem sessionsweresupervisedbyDr.JoyJacob,Dr.SebastianGeorgeandotherlecturers. The5thS.E.R.C.Schoolinthissequence,titled“SpecialFunctionsandFunctions of Matrix Argument: Recent Advances and Applications in Stochastic Processes, Statistics, Wavelet Analysis and Astrophysics” was conducted by CMS at its Pala Campus from 23rd April to 25th May 2007. The lecture notes for the 5th School were assembled by November 2006 and were printed and distributed as Publica- tionNo.34inthePublicationsSeriesofCMS.Themainlectureswerescheduledto begivenbyDr.HansJ.Haubold(OfficeofOuterSpaceAffairsoftheUnitedNa- tions,Austria),Dr.H.M.Srivastava(UniversityofVictoria,Canada),Dr.R.Y.Denis (Gorakhpur University, India), Dr.R.K. Saxena (Jodhpur University, India), Dr. S. Bhargava(MysoreUniversity,India),Dr.D.V.Pai(IITBombay,India),Dr.Yageen Thomas(UniversityofKerala,India),Dr.K.K.Jose(MahatmaGandhiUniversity, India), Dr. J.J. Xu (China/Canada), Dr. K. Jayakumar (Calicut University, India) andDr.A.M.Mathai(Canada/India).ButDr.H.M.Srivastava,Dr.S.Bhargavaand Dr.Xucouldnotreachthevenueontimeduetounexpectedemergencies.Supple- mentarylecturesweregivenbyDr.A.SukumaranNair(Chairman,CMS),Dr.R.N. Pillai (former Head, Department of Statistics, University of Kerala, India), Dr. N. Mukunda(IISc,Bangalore,India)andtheproblemsessionsweresupervisedbyDr. Sebastian George, Dr. Joy Jacob, Dr. Seemon Thomas and the lecturers. For the 2005,2006and2007Schools,Dr.A.M.MathaiwastheDirectorandDr.K.K.Jose wastheCo-DirectoroftheSchools. TheparticipantsfortheS.E.R.C.Schoolsareselectedonall-Indiabasis.Allthe expenses of the selected candidates, total number of seats is 30, including travel, accommodation,food,lecturematerials,localtravelsetcaremetbytheDST,Delhi, GovernmentofIndia.Foreignparticipationisallowedundertheconditionsthatthe Preface vii participantsmustcomewiththeirownreturnairticketsandmustattendallthelec- tures,andproblemsessionsandtakealltheexaminationsandtestsfromthebegin- ningtilltheend.Alltheirlocalexpensesaremetandlecturematerialsareprovided bytheSchools.ThereisnofeeforattendingtheSchools.TheSchoolsaremainly research orientation courses aimed at young faculty members in colleges and uni- versitiesacrossIndia,below35yearsofage,andfreshgraduateswithM.Sc,M.Phil, Ph.Ddegreesbelow30yearsofage,inMathematics/Statistics/TheoreticalPhysics. In most of the Indian universities rigid compartmentalization is the order and as a resultaM.Scgraduateinmathematicsmaynothaveeventakenaverybasiccourse inprobabilityandstatistics.Eventhoughbasicdifferentialandintegralcalculusand matrix theory are required subjects for statistics and physics students, these stu- dentsmayhaveforgottenthesesubjectsbecausetheyteachthecompartmentalized topics in their own areas and do not usually do research even in their narrow ar- easoftheirownfields.Formaintainingtheirjobsandgettingregularincrementsin their salaries and all other monetary and other benefits, research work and further reading and learning process are not required of them. As a result, the quality of teachingandtheinformationpassedontothestudentsgodownfromyeartoyear. Inordertoremedythissituationalittlebit,S.E.R.C.Schoolsinvariousareaswere startedbyDST,Delhi.Dr.A.M.MathaiwasaskedtorunS.E.R.C.Schoolsinmath- ematics. For taking up a challenging research problem in any applied area a good backgroundinbasicmathematics,probabilityandstatisticsisrequired.Inorderto givethebasicideasinprobabilityandstatisticsandtobringanumberoftopicsin the area of Special Functions and Functions of Matrix Argument and their appli- cations to the current research level, these S.E.R.C. Schools on Special Functions wereestablished.ThecurrentsequenceoffiveSchoolsreallyachievedtheaimand almost all the participants in the first four Schools have become research oriented towardsacareerinresearchandteaching,andtheparticipantsofthe5thSchoolare alsoexpectedtofollowthesamefootstepsoftheirpredecessors. Chapter 1 introduces elementary classical special functions. Gamma, beta, psi, zetafunctions,hypergeometricfunctionsandtheassociatedspecialfunctions,gen- eralizationstoMeijer’sGandFox’sH-functionsarealsoexaminedhere.Discussion isconfinedtobasicpropertiesandsomeapplications.Introductiontostatisticaldis- tribution theory is given here. Some recent extensions of Dirichlet integrals and Dirichlet densities are also given. A glimpse into multivariable special functions suchasAppell’sfunctionsandLauricellafunctionsisalsogiven.Specialfunctions assolutionsofdifferentialequationsarealsoexaminedhere. Chapter 2 is devoted to fractional calculus. Fractional integrals and fractional derivativesofvariouskindsarediscussedhere.Thentheirapplicationstoreaction- diffusion problems in physics, input-output analysis and Mittag-Leffler stochas- tic processes are examined here. Chapter 3 deals with q-hypergeometric or basic hypergeometric functions and Chapter 4 goes into basic hypergeometric functions andRamanujan’sworkonellipticandthetafunctions.Chapter5examinesthetopic ofSpecialFunctionsandLieGroups. viii Preface Chapters 6 to 9 are devoted to applications of Special Functions to various ar- eas. Applications to stochastic processes, geometric infinite divisibility of random variables,Mittag-Lefflerprocesses,α-Laplaceprocesses,densityestimation,order statisticsandvariousastrophysicsproblems,aredealtwithinthesechapters. Chapter10isdevotedtoWaveletAnalysis.Anintroductiontowaveletanalysisis givenhere.Chapter11dealswiththeJacobiansofmatrixtransformations.Various typesofmatrixtransformationsandtheassociatedJacobiansaregivenhere.Chapter 12isdevotedtothediscussionoffunctionsofmatrixargument.Onlytherealcase isconsideredhere.Functionsofmatrixargumentandthepathwaymodels,recently introducedbyMathai(2005),arealsodiscussedhere,alongwiththeirapplications tovariousareas. InalltheS.E.R.C.SchoolsconductedundertheDirectorshipofDr.A.M.Mathai aseriouseffortismadesothattheparticipantsabsorbthematerialscoveredinthe Schools. The classes started at 8.30 am and went until 6.00 pm. The first lecture of 08.30 to 10.30 was followed by a problem session from 10.30 to 13.00 hrs on the materials covered in the first lecture. The second lecture of the day was from 14.00 to 16.00 hrs followed by problem session from 16.00 to 18.00 hrs. At the end of every week a written examination was conducted, followed by a personal interview of each participant by the lecturer of that week in the form of an oral examination.Cumulativegradesofsuchweeklyexaminationsappearedinthefinal certificates distributed to them. The main aim was to inculcate in them a habit of long and sustained hard work, which would help them in their careers whatever they may be. During the first week the participants, especially the teachers from collegesanduniversities,founditdifficulttoadjusttotheroutineoflonghoursof hard work but starting from the second week, in all the Schools, the participants started enjoying, especially the problem sessions, because for the first time, they started understanding and appreciating the meanings and significance of theorems thattheylearntormemorizedwhentheywerestudents. Thelecturenotesarewrittenupinastyleforself-study.Eachtopicisdeveloped fromfirstprincipleswithlotsofworkedexamplesandexercises.Hencethematerial inthisbookcanbeusedforself-studyandwillhelpanyonetounderstandthebasic ideasintheareaofSpecialFunctionsandFunctionsofMatrixArgumentandthey willbeabletomakeuseoftheseresultsintheirownproblemsinappliedareas,espe- ciallyinStatistics,PhysicsandEngineeringproblems.Applicationsinvariousareas areillustratedinthisbook.Insightsintorecentdevelopmentsintheapplicationsof fractionalcalculus,inthedevelopmentsinvariousothertopicsarealsogiveninthe booksothatthereaderswhoareinterestedinanyofthetopicsdiscussedinthebook candirectlygointoaresearchprobleminthetopics. SeveralpeoplehavecontributedenormouslyforthesuccessoftheS.E.R.C.Schools and in making the publications of four Lecture Notes and this final publication of summarized lecture notes possible. Dr. B.D. Acharya, Advisor to Government of IndiaandDr.AshokK.SinghoftheMathematicalSciencesDivisionofDST,New Delhi,arethedrivingforcebehindthere-energizedmathematicalactivitiesinIndia Preface ix now. They were kind enough to pursue the matter and get the funds released for runningtheSchoolsaswellasforthepreparationofvariouspublications,including this one. Since the basic materials for this publication are supplied by various lecturersintheformoftheirlecturenotes,therewillbesomeoverlaps.Veryobvious inconsistenciesareremovedbutsomeoverlappingmaterialsarelefttheretomakethe discussionsself-contained.Dr.R.K.Saxena,Dr.S.Bhargava,Dr.H.J.Haubold,Dr. P.N.Rathie,Dr.K.K.Jose,Dr.K.Jayakumar,Dr.H.L.Manocha,Dr.R.Jagannathan, Dr. K. Srinivasa Rao, Dr. K.S.S. Nambooripad, Dr. Serge B. Provost, Dr. Yageen Thomas, Dr. D.V. Pai, and Dr. A.M. Mathai are thanked for making their notes available in advance for the S.E.R.C. Schools. Most of the material was typeset at CMS office by Miss K.H. Soby, Dr. Joy Jacob, Seemon Thomas, Dr. Sebastian George, Dr. K.K. Jose and Dr. A.M. Mathai. Part of the material was typeset by Barbara Haubold of the United Nations, Vienna Office, fully free of charge as a voluntary service to the Schools. Notes and programs for a series of lectures on MatlabweresuppliedtoCMSbyAlexanderHauboldofColumbiaUniversity,USA. Thosenotesarenotincludedinthisbooktokeepthematerialswithinthefocusof thebook.Dr.SebastianGeorge,Dr.JoyJacob,Dr.SeemonThomas,K.M.Kurian, Jaisymol Thomas and Ashly P. Jose, who spent a lot of time in running problem sessions,inrunningseparatesessionsontheuseofstatisticalpackagesandMaple program,useofLATEXetcandforcheckingthetypedmaterials,arethanked.CMS would like to thank each and every one who helped to make S.E.R.C. Schools on SpecialFunctionsagrandsuccessandwhohelpedtomakethispublicationpossible. The authors would like to express their sincere thanks to the Department of ScienceandTechnology,GovernmentofIndia,NewDelhi,India,forthefinancial assistanceundertheprojectNo.SR/S4/MS:287/05titled“BuildingupaCoreGroup of Researchers/Faculty and Facilities at CMS, Trivandrum and Pala Campuses”, whichenabledthecollaborationonthisbookprojectpossible.Weexpressoursin- ceregratitudetotheManagementandespeciallytoDr.MathewJohnK.(Principal), St. Thomas College Pala, Kerala, India, for providing all facilities in the College duringthepreparationofthisbook. Pala,Kerala,India A.M.Mathai 1st February,2007 H.J.Haubold Glossary of Symbols (b) PochhammerSymbol Notation1.0.1 1 r n(cid:1)!(cid:2) factorialnornfactorial Notation1.0.2 2 n numberofcombinationsofn Notation1.0.3 2 r takenratatime R(·) realpartof(·) Notation1.1.1 3 Γ(z) gammafunction Notation1.1.1 3 γ Euler’sconstant Notation1.1.2 3 π pi,mathematicalconstant Definition1.1.6 4 ∧ wedgeorskewsymmetricproduct Notation1.1.3 6 J Jacobian Definition1.1.7 6 (a) B (x) generalizedBernoullipolynomial Noation1.1.4 9 k B (x) Bernoulipolynomialoforderk Notation1.1.4 9 k B Bernoullinumberoforderk Notation1.1.4 9 k ψ(·) psifunction Section1.2 12 ζ(ρ) Riemannzetafunction Section1.2.1 12 ζ(ρ,a) generalizedzetafunction Section1.2.1 12 M (s) Mellintransformof f Section1.3 14 f L (t) Laplacetransformof f Section1.3.2 17 f F (t) Fouriertransformof f Section1.3.2 18 f P(A) probabilityoftheeventA Section1.4 19 E(·) expectedvalue Notation1.5.1 28 Var(·) varianceof(·) Definition1.5.2 29 Cov(·) covariance Exercises1.5.7 34 B(α,β) betafunction Section1.6 34 D(α,...,α; 1 k α ) Dirichletfunction Notation1.6.2 37 k+1 F (·) hypergeometricseries Section1.7 42 p q γ(α;b) incompletegammafunction Section1.7 43 Γ(α;b) incompletegammafunction Section1.7 43 b(α,β;t) incompletebetafunction Section1.7 44 B(α,β;t) incompletebetafunction Section1.7 44 xi xii GlossaryofSymbols Gm,n[·] G-function Section1.8 49 p,q Jν(·) Besselfunction Exercise1.8 54 Iν(·) Besselfunction Exercises1.8 54 Hm,n(·) H-function Section1.9 54 p,q Eα,β(·) Mittag-Lefflerfunction Section1.9 57 f (·),f (·), Lauricellafunction Section1.10 58 A B f (·),f (·) Lauricellafunction Section1.10 58,59 C D (1) (2) Hν (x),Hν (x) Besselfunctionofthethirdkind Section1.11.6 73 (α) L (x) Laguerrepolynomial Section1.11.7 74 n P (x) Legendrepolynomial Section1.11.8 74 m Eα(x),Eα,β(x) Mittag-Lefflerfunctions Section2.1 79 h (z,n) hyperbolicfunctionofordern Definition2.1.3 80 r K (z,n) trigonometricfunctionofordern Definition2.1.4 80 r erf,erfc errorfunctions Definition2.1.5 80 E(ν,α) Mellin-Rossfunction Definition2.1.6 81 t Rα(β,t) Robotov’sfunction Definition2.1.7 81 F(s)=L{f(t);s} Laplacetransform Notation2.2.1 85 L−1{f(s);t} inverseLaplacetransform Notation2.2.2 85 δ E (x) generalizedMittag-Leffler Notation2.3.1 91 β,γ function In, D−n Riemann-Liouvillefractional a x a x integrals Notation2.4.1 98 −α α D , I Riemann-Liouvilleleft-sided a x a+ fractionalintegrals Definition2.4.2 99 α −α I , D Riemann-Liouvilleright-sided x b x b fractionalintegrals Definition2.4.3 99 xW∞α, xI∞α: Weylintegraloforderα Notation2.4.4 103 α α xD∞, D−: Weylfractionalderivatives Notation2.4.5 103 α α −∞Wx ,I+ Weylintegrals Notation2.4.6 104 α C D Caputoderivative Section2.4.9 108 a x 0 m{f(x);s},f∗(s) Mellintransform Section2.5 109 m−1{f∗(s);x} inverseMellintransform Section2.5 109 I[f(x)], Koberoperatorofthefirstkind Notation2.6.1 112 I[α,η: f(x)] Koberoperatorofthefirstkind Notation2.6.1 112 I(α,η)f(x)Eα,η f Koberoperatorofthefirstkind Notation2.6.1 112 0,x In,α f Koberoperatorofthefirstkind Notation2.6.1 112 x R[f(x)] Koberoperator,secondkind Notation2.6.2 112 R[α,ζ: f(x)] Koberoperator,secondkind Notation2.6.2 112 R(α,ζ)f(x) Koberoperator,secondkind Notation2.6.2 112 α,ζ ζ,α Kx,∞ f,Kx f Koberoperator,secondkind Notation2.6.2 112 I[·],K[·] generalizedKoberoperators Section2.7 114 Iα,β,η(·) fractionalintegral/derivative Definition2.7.7 115 0,x ofthefirstkind