Table Of ContentTutorials, 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
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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
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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
