Tutorials, Schools, and Workshops in the Mathematical Sciences Mama Foupouagnigni Wolfram Koepf Editors Orthogonal Polynomials 2nd AIMS-Volkswagen Stiftung Workshop, Douala, Cameroon, 5-12 October, 2018 Tutorials, Schools, and Workshops in the Mathematical Sciences Thisserieswillserveasaresourceforthepublicationofresultsanddevelopments presentedatsummerorwinterschools,workshops,tutorials,andseminars.Written in an informal and accessible style, they present important and emerging topics in scientific research for PhD students and researchers. Filling a gap between traditional lecture notes, proceedings, and standard textbooks, the titles included inTSWMSpresentmaterialfromtheforefrontofresearch. Moreinformationaboutthisseriesathttp://www.springer.com/series/15641 Mama Foupouagnigni • Wolfram Koepf Editors Orthogonal Polynomials 2nd AIMS-Volkswagen Stiftung Workshop, Douala, Cameroon, 5-12 October, 2018 Editors MamaFoupouagnigni WolframKoepf UniversityofYaoundéI InstituteforMathematics Yaoundé,Cameroon UniversityofKassel Kassel,Germany AfricanInstituteforMathematicalSciences Limbe,Cameroon ISSN2522-0969 ISSN2522-0977 (electronic) Tutorials,Schools,andWorkshopsintheMathematicalSciences ISBN978-3-030-36743-5 ISBN978-3-030-36744-2 (eBook) https://doi.org/10.1007/978-3-030-36744-2 MathematicsSubjectClassification(2010):33C,33D,33F ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword In July 2015, following an invitation from the African Institute for Mathematical Sciences (AIMS) Global Secretariat, the two authors of this foreword decided to applyforfundingtotheVolkswagenFoundation’sSymposiaandSummerSchools initiative to support the organization of two workshops planned to take place in Cameroon. AIMS is a pan-African network of centres of excellence for post-graduate training,researchandpublicengagementinmathematicalsciences.AIMSenables Africa’sbrighteststudentstobecomeinnovatorsthatpropelscientific, educational andeconomicself-sufficiency. AIMS-Cameroon is the fourth Center of Excellence of the AIMS Network created in 2013 after AIMS South Africa, AIMS Senegal and AIMS Ghana, followed by the creation of AIMS Tanzania and AIMS Rwanda. It is located in Limbe in Cameroon, a country from the Central Africa sub-region, well known as Africa in miniature due to its diverse landscapes that represent the continent’s majorclimaticzones.Amongthese,wecanmentionthatwhiteandblackbeaches, mountainousareas,tropicalrainforests,savannagrasslandsandsparse desertsare found in this country. As illustration, the Mount Cameroon, located in Buea near Limbe,isthehighestpointincentralandwestAfricawhileDebundschaalsolocated nearLimbeisthesixthwettestplaceintheworld. Of course, it took quite a long time to develop this idea and make it concrete: We had to propose programmes for the two planned meetings onIntroductionto ComputerAlgebraandApplicationsandonIntroductiontoOrthogonalPolynomials andApplications. We had also to invite renowned internationalexperts to present plenarylectureson the currentstate of the artin their domains,andwe were very happyabouttheiracceptance.Wehadtowritethecorrespondingproposalandtoget approvalfrom the reviewers and finally from Volkswagen Foundation.In the first round,somereviewersofourproposalhadrecommendedustoaddsomeadditional actualmajorresearchfieldsinorthogonalpolynomialswhichwedid. InMarch2017,wefortunatelyreceivedthepositivegrantletterfromVolkswagen Foundationandcouldstartthefinalplanning.ThefirstworkshoponIntroductionto ComputerAlgebraandApplicationstookplaceonOctober6–13,2017,whereasthe v vi Foreword secondworkshoponIntroductiontoOrthogonalPolynomialsandApplicationswas scheduledforOctober5–12,2018,bothtakingplaceintheHotelPrincedeGalles in Douala, the economic capital of Cameroon. They were hosted by the African InstituteforMathematicalSciences,Cameroon(https://www.aims-cameroon.org/), and were co-organizedtogetherwith the University of Kassel in Germany (http:// www.mathematik.uni-kassel.de/~koepf/).All details aboutthe two workshopscan be found on the web domain http://www.aims-volkswagen-workshops.org/of the meetingswhichisstillactive. These Proceedingscontain the results of the second workshop. This workshop onIntroductiontoOrthogonalPolynomialsandApplicationswasaimedgloballyat promotingcapacity building in terms of research and training in orthogonalpoly- nomialsandapplications,discussionsanddevelopmentofnewideas,development andenhancementofnetworkingincludingsouth–southcooperation. Theworkshopbroughttogether60participantsfrom18African,7Europeanand 2NorthAmericancountriesincluding19plenaryspeakersandtrainerswhoareall expertsintheirvariousdomains.Intotal,about50plenarytalks,tutorials,training sessions and contributed talks were delivered during the preliminary workshop (October5–7,2018)andtheworkshop(October8–12,2018). To announce the workshop, we designed a nice and informative web page at http://www.aims-volkswagen-workshops.org/anda linktothefirstworkshopweb site was included. Also, a Facebook event page was installed. On this web page, informationaboutthe objectivesand expectationsof the workshop,the organizers andthefundingpartners,theplenaryspeakers(CVandphoto),programmeschedule andabstractsoflecturesandtutorialscouldbedownloaded,alsobythoseinterested researchersthatwecouldnotinvite.Also,wehaveputonthewebsite andspread by email all over Africa the call for application. Those interested to apply had to fillintheonlineapplicationformfromourwebsite.Asaresult,wehavereceived 130applicationswithinabout1month.Manyofthese applicationswereexcellent sothatwecouldfinallyinvite25AfricansfromoutsideCameroonfrom18different countries and 16 Cameroonian researchers. 10 of our African participants were female.The6Africanparticipantswiththebestproposalsforatalkwereselected topresenttheirresearchattheworkshop. Theworkshopevaluationbytheparticipantswasverypositive,andtheworkshop which enabled active interactions between the participants was a great success, enablingtheachievementofthestatedobjectives! SincewedidnotexpectpriorknowledgeabouttheworkshoptopicbytheAfrican participants, the preliminary workshop was giving a formal introduction to the field of orthogonal polynomials. This was possible through the great help of a groupoffiveformerCameroonianPhDstudentswhoallwrotetheirdissertations— supervisedbythetwoworkshoporganizers—inthefieldofOrthogonalPolynomials and Special Functions and whom we are very grateful for their help: Maurice KenfackNangho,SalifouMboutngam,MerlinMouafoWouodjie,PatrickNjionou Sadjang and Daniel D. Tcheutia. Finally, the talks given by the two organizers andthe oneof Aletta Jooste fromthe Universityof Pretoriain SouthAfrica (who couldnotattendduetolastminutehealthconstraints)givenbyDanielD.Tcheutia, Foreword vii complemented the lecturers of the preliminary workshop. All those contributions arecontainedintheseProceedings. The main workshop was aimed at introducing concepts of modern and actual research topics in orthogonal polynomials: Multiple Orthogonal Polynomials, Orthogonal Polynomials and Painlevé Equations, Orthogonal Polynomials and RandomMatrices,OrthogonalPolynomialsinSobolevSpaces,MatrixPolynomials, ZerosofOrthogonalPolynomials,ComputerAlgebraandOrthogonalPolynomials, MultivariateOrthogonalPolynomials,Askey–WilsonScheme. Based ontheconceptoftheworkshop,we havedividedthese Proceedingsinto twoparts.PartIgivesanIntroductiontoOrthogonalPolynomialsbasedonthetalks given in the preliminary workshop. They are organized in their logical structure. Part II presents the remaining lectures on ActualResearchTopicsinOrthogonal PolynomialsandApplications. They are organized in alphabetical order of their authors.Anorderingbytheirtopicswasnotpossiblesincesomeauthorspreferred towriteonearticleaboutseveraltopics. WehopethattheseProceedingswillnotonlygiveaverygoodintroductioninto thestateoftheactualresearchinorthogonalpolynomialsandapplications,butalso helpthoseinterestedinorthogonalpolynomialswithoutpriorknowledgetoembark intothisinterestingfieldofresearch. Acknowledgements WewouldliketothankVolkswagenFoundationfortheirgreatsupport!Withoutthis generousgrant (Symposia and Summer Schools, Wissen für Morgen,AZ 93 000, http://portal.volkswagenstiftung.de/search/projectDetails.do?ref=93000)thiswork- shop clearly would not have been possible. Furthermore, we would like to thank Clemens Heine from Birkhäuser who had the idea to collect this volume after visiting our web site. Our thanks go also to the plenary speakers, the trainers and Ms Nathalie Diane Wandji Nanda for their decisive contributions to the success ofthisworkshop.We arealsodelightedtoacknowledgeandvaluethetremendous effortsoftheAIMSGlobalNetworkinbuildingthecapacityofthenextgeneration of African scientists and innovators. We would also like to thank the Alexander von Humboldt Foundation for continuously supporting the academic cooperation betweenthetwoauthors.Thiscollaborationhasmadepossibletheorganizationof thetwoVolkswagenworkshopsfromwherethecurrentproceedingsemerged. Inthesameline,MamaFoupouagnigniisverypleasedtoacknowledgeherewith bigthanksthevaluableanddecisivecontributionofWolframKoepftothecapacity building of the Cameroon mathematicalcommunity by means of supervision, co- supervision or mentoring, which has led to the successful completion of at least 7 PhD theses in mathematics (Mama Foupouagnigni,Etienne Nana Chiadjeu and theabovelistedfiveformerCameroonianPhDstudents,BertrandTeguiaTabuguia (PhD thesis almost completed)); and at least three German Habilitation theses in mathematics (Mama Foupouagnigni, Jean Sire Eyebe Fouda and Daniel Duviol viii Foreword Tcheutia, Patrick Sadjang Njionou (Habilitation thesis almost completed)). His wife Angelika Wolf is also acknowledged for having continuously supported this academiccooperationwhichfirststartedin1997asstudent–supervisorrelationship, thenmigratedtocollaborationamongtwoHumboldtians.Ithasnowdevelopedinto an active institutional academic cooperation combined with sustained friendship and extended family ties, connecting Cameroon and Germany. Such academic cooperationand culturaldialogue are in line with the objectivesof the Alexander von Humboldt Foundation which is to promote academic cooperation between excellent scientists and scholars from abroad and from Germany, as an interme- diary organization for German foreign cultural and educational policy promoting internationalculturaldialogueandacademicexchange. Limbe,Cameroon MamaFoupouagnigni Kassel,Germany WolframKoepf August2019 Contents PartI IntroductiontoOrthogonalPolynomials AnIntroductiontoOrthogonalPolynomials.................................. 3 MamaFoupouagnigni ClassicalContinuousOrthogonalPolynomials............................... 25 SalifouMboutngam Generating Functions and Hypergeometric Representations ofClassicalContinuousOrthogonalPolynomials............................ 45 MauriceKenfackNangho PropertiesandApplicationsoftheZerosofClassicalContinuous OrthogonalPolynomials......................................................... 61 A.S.Jooste Inversion,MultiplicationandConnectionFormulaeofClassical ContinuousOrthogonalPolynomials .......................................... 69 DanielDuviolTcheutia ClassicalOrthogonalPolynomialsofaDiscreteandaq-Discrete Variable ........................................................................... 85 PatrickNjionouSadjang ComputerAlgebra,PowerSeriesandSummation........................... 113 WolframKoepf OntheSolutionsofHolonomicThird-OrderLinearIrreducible DifferentialEquationsinTermsofHypergeometricFunctions............. 137 MerlinMouafoWouodjié TheGammaFunction ........................................................... 155 DanielDuviolTcheutia ix