Lecture Notes inMathematicsEdited by A Dold and B. Eckmann
1171
Polynomes Orthogonauxet ApplicationsProceedings of the Laguerre Symposiumheld at Bar-le-Duc, October 15-18, 1984
Edite parC. Brezinski, A. Draux, A.P. Magnus, P. Maroni et A. Ronveaux
Springer-VerlagBerlin Heidelberg NewYork Tokyo
Editeurs
ClaudeBrezinskiAndreDrauxUniversite de Lille 1, U.E.R. I.E.E.A. Informatique59655 Villeneuve d'Ascq Cedex, France
Alphonse P. MagnusInstitutde Mathematique, U.C.L.Chemin du Cyclotron 2, 1348 Louvain-Ia-Neuve, Belgique
Pascal MaroniUniversite Pierreet MarieCurieU.E.R. Analyse, Probabilites et Appl.4 PlaceJussieu, 75252 ParisCedex05, France
AndreRonveauxDepartement de Physique, FacultesUniversitaires N.D. de la Paix61 rue de Bruxelles, 5000 Namur, Belgique
Mathematics Subject Classification (1980): 30E 10, 41A 10, 41A21, 42C
ISBN3-540-16059-0Springer-Verlag Berlin Heidelberg New York TokyoISBN 0-387-16059-0Springer-Verlag New York Heidelberg BerlinTokyo
This work is subject to copyright. All rights are reserved, whether the whole or part of the materialis concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,reproduction by photocopying machine or similar means, and storage in data banks. Under§ 54 of the German Copyright Law where copies are made for other than private use, a fee ispayable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.2146/3140-543210
Edmon.d LagueJUte..
PRtFACE
Depuis quelque temps un groupe de travail sur les polynOmesorthogonaux reunissait les organisateurs de ce Symposium lorsque, enNovembre 1982, nous re¥umes tous une lettre d'Andre Ronveaux nous signalant qu'on feterait en 1984 Ie 150ieme anniversaire de la naissancede Laguerre et nous proposant de nous associer pour organiser, a cetteoccasion, un congres international sur les polynOmes orthogonaux etleurs applications. Andre devait commencer a desesperer d'avoir unereponse lorsque, lors d'une reunion ulterieure de notre groupe de travail, l'idee revint a la discussion et la decision fut prise.
Les premiers problemes a regler concernaient Ie financementet Ie lieu. Laguerre est ne et mort a BarLeDuc, Ie lieu s'imposaitpresque de luimeme. Nous primes done contact avec la municipalite.L'accueil qui nous fut reserve depassa de beaucoup nos previsions lesplus optimistes. Non seulement une subvention importante nous futaccordee mais Ie personnel de la mairie fut mis a notre dispositionpour nous aider a la preparation du congres. Enfin la municipaliteprit a sa charge, materielle et financiere, tous les problemes locauxcomme Ie centre des conferences, les pauses, les polycopies des resumes, les taxis, les distractions, ... La liste de ce que nous devonsa Monsieur Bernard, DeputeMaire de BarLeDuc, et a ses collaborateursest trop longue pour avoir sa place ici, mais il est certain que ceSymposium n'aurait pas pu avoir lieu sans leur aide et leur devouement.Si nous pouvons parler de reussite, c'est en partie a eux quenous la devons et nous tenons a les en tous tres chaleureusement.
Bien que Ie programme scientifique ait ete tres charge puisque plus de soixantedix communications furent presentees par la centaine de participants venus de seize pays, Ie cOte culturel n'avaitpas ete oublie. Au cours de la premiere matinee de travail, Ie Professeur J. Dieudonne, membre de l'Academie des Sciences, rappela la vieet l'oeuvre de Laguerre devant un public compose du Prefet, du DeputeMaire, des personnalites civiles et militaires de la region, descongressistes et des eleves des classes terminales du lycee. Ensuiteles participants furent convies au bapteme d'un groupe scolaire du nomde Laguerre. Apres un discours de M. Bernard, DeputeMaire, la plaqueen l'honneur de Laguerre fut devoilee par Ie Professeur Dieudonne. Lescongressistes eurent egalement l'occasion de visiter la vieille villede BarLeDuc qui presente un tres bel ensemble de maisons renaissance,d'assister a un concert de jazz et de prendre part a un banquet tresanime et cordial, preside par Monsieur Ie Prefet.
Nous tenons egalement a exprimer notre reconnaissance auxdivers organismes qui nous ont apporte leur aide financiere : CentreNational de la Recherche Scientifique, Societe Mathematique de France,College de Mathematiques Appliquees de l'AFCET et Compagnie Bull.
Nous remercions les editeurs BirkhauserVerlag et SpringerVerlag pour avoir apporte leur concours a l'organisation de l'exposition de livres et J. Labelle de l'Universite du Quebec a Montreal qui
VI
nous a fourni les tableaux d'Askey sur les polyn6mes orthogonaux.Enfin au nom de tous les participants nous voulons dire a nos h6tessesMuriel Colombo, Any Pibarot et Liliane Ruprecht combien nous avonsapprecie leur efficacite souriante. Nous n'oublions pas non plus SaidBelmehdi pour son aide precieuse.
Nous esperons que ce Symposium, qui fut en fait Ie premierCongres International entierement consacre aux polyn6mes orthogonauxet a leurs applications, sera suivi de beaucoup d'autres. C'est Ievoeu que nous formulons.
C. BREZINSKI
A. DRAUX
A. MAGNUS
P. MARONI
A. RONVEAUX
COHSRES SUERRE
TABLE DES MATIERES
vPREFACE
LISTE DES PARTICIPANTS xv
EDMOND NICOLAS LAGUERRE par C. Brezinski XXI
LAGUERRE Alii> ORTHOGONAL POLYNlJotIALS IN 1984 par A.P. Magnus et A. Ronveaux XXVII
TABLEAU Df ASKEY par J. Labelle XXXVI
I. CONFERENCIERS INVITES
DIEUDONNE J.,
HAHN W.,
Fractions continuees et polynomes orthogonaux dans
l'oeuvre de E.N. LAGUERRE.
Uber Orthogonalpolynome, die linearen Funktional
gleichungen genugen.
16
ANDREWS G.E., ASKEY R., Classical orthogonal polynomials. 36
GAUTSCHI W., Some new applications of orthogonal polynomials. 63
II. CONFERENCIERS OU CONTRIBUTEURS *
1. CONCEPTS D'ORTHOGONALITE
DE BRUIN M.G.,
DRAUX A.,
Simultaneous Pade approximation and orthogonality.
Orthogonal polynomials with respect to a linear
functional lacunary of order S + 1 in a noncommu
tative algebra.
74
84
ISERLES A., N0RSETT S.P., Biorthogonal polynomials. 92
KOWALSKI M.A.,
VIII
Algebraic characterization of orthogonality in the
space of polynomials. 101
2. COHBINATOIRE ET GRAPHES
BERGERON F., Une approche combinatoire de la methode de Weisner. 111
de SAINTE-CATHERINE M., VIENNOT G., Combinatorial interpretation of inte-
grals of products of Hermite, Laguerre and Tchebycheff 120
polynomials.
STREHL V.,
VIENNOT G.,
Polyn6mes d'Hermite generalises et identites de SZEGO-
une version combinatoire.
Combinatorial theory for general orthogonal polynomials
with extensions and applications.
129
139
3. ESPACES FONCTIONNELS
ALFARO P., ALFARO M., GUADALUPE J.J., VIGIL L.,
Correspondance entre suites de polyn6mes orthogonaux et
fonctions de la boule unite de H:(B). 158
DE GRAAF J.,
KOORNWINDER T.H.,
MARONI P.,
4. PLAN COIfPLEXE
Two spaces of generalized functions based on harmonic
polynomials.
Special orthogonal polynomial systems mapped onto each
other by the FOURIER-JACOBI transform.
Sur quelques espaces de distributions qui sont des for-
mes lineaires sur llespace vectoriel des polyn6mes.
164
174
184
GARCIA-LAZARO P., MARCELLAN F., Christoffel formulas for N-Kernels asso-
ciated to Jordan arcs. 195
IX
GUADALUPE J.J., REZOLA L., Closure of analytic polynomials in weighted
Jordan curves.
MARCELLAN F., MORAL L., Minimal recurrence formulas for orthogonal poly
nomials on Bernoulli's lemniscate.
5. I!fBSURES
204
211
LUBINSKY D.S.,
NEVAI P.,
PASZKOWSKI 5.,
ULLMAN J.L.,
6. ZBROS
Even entire functions absolutely monotone in [0, 00)
and weights on the whole real line.
Extensions of Szego's theory of orthogonal polynomials.
Sur des transformations d'une fonction de poids.
Orthogonal polynomials for general measuresII.
221
230
239
247
ALVAREZ M., SANSIGRE G., On polynomials with interlacing zeros. 255
GILEWICZ J., LEOPOLD Eo, On the sharpness of results in the theory of
location of zeros of polynomials defined by three term
recurrence relations. 259
LAFORGIA A.,
RUNCKEL H.J.,
SABLONNIERE P.,
Monotonicity properties for the zeros of orthogonal
polynomials and Bessel functions.
Zeros of complex orthogonal polynomials.
Sur les zeros des splines orthogonales.
267
278
283
VINUESA J., GUADALUPE R.,
Zeros extremaux de polynomes orthogonaux.
7. APPROXIlfATIONS
291
DERIENNIC M.M., Polynomes de Bernstein modifies sur un simplexe T.e
de R. Problemes des moments.
296
x
KANO T., On the size of some trigonometric polynomials. 302
LOPEZ LAGOMASINO G.,Survey on multipoint Pade approximation to Markov
type meromorphic functions and asymptotic proper-
ties of the orthogonal polynomials generated by them. 309
PASZKOWSKI S., Une relation entre les series de Jacobi et l'appro-
ximation de Pade. 317
* STAHL H., On the divergence of certain Pade approximant and the
behaviour of the associated orthogonal polynomials. 321
8. FAMILLES SPECIllLES
DURAND L.,
GROSJEAN C.C.,
Lagrangian differentiation, Gauss-Jacobi integration,
and Sturm-Liouville eigenvalue problems.
Construction and properties of two sequences of ortho-
gonal polynomials and the infinitely many, recursively
generated sequences of associated orthogonal polyno-
mials, directly related to Mathieu's differential
equation and functions - Part I -
331
340
HENDRIKSEN E., van ROSSUM H., Semi-classical orthogonal polynomials. 354
MAGNUS A.P.,
McCABE J.,
MEIJER H.G.,
WIMP J.,
A proof of Freud's conjecture about the orthogonal
polynomials related to Ixl P exp (_x2m), for integer m.
Some remarks on a result of Laguerre concerning con-
tinued fraction solutions of first order linear diffe-
rential equations.
Asymptotic expansion of Jacobi polynomials.
Representation theorems for solutions of the heat
equation and a new method for obtaining expansions
in Laguerre and Hermite Polynomials.
362
373
380
390
XI
9. ANALYSE NUlfBRIQUB
DEVILLE M., MUND E.,On a mixed one step/Chebyshev pseudospectral tech
nique for the integration of parabolic problems using
finite element preconditioning. 399
GONZALEZ P., CASASUS L., Two points Pade type approximants for
Stieltjes functions. 408
MASON J.C.,
*MONSION M.,
Nearminimax approximation and telescoping procedures
based on Laguerre and Hermite polynomials.
Application des polyn6mes orthogonaux de Laguerre al'identification des systemes nonlineaires.
419
426
*NAMASIVAYAM S., ORTIZ E.L., On figures generated by normalized Tau
approximation error curves. 435
NEX C.M.M.,
SHAMIR T.,
TEMME N.M.,
VIANO G.A.,
10. APPLICATIONS
BLACHER R.,
Gausslike integration with preassigned nodes and
analytic extensions of continued fractions.
Orthogonal polynomials and the partial realization
problem.
A class of polynomials related to those of Laguerre.
Numerical inversion of the Laplace transform by the
use of Pollaczek polynomials.
Coefficients de correlation d'ordre (I, J) et varian
ces d'ordre I.
442
451
459
465
475
GASPARD J.P., LAMBIN P., Generalized moments application to solidstate
physics. 486
XII
KIBLER M., NEGADI T., RONVEAUX A., The Kustaanheimo-Stiefel transfor-
mation and certain special functions.
LAW A.G., SLEDD M.B., A non classical, orthogonal polynomial family.
497
506
LINGAPPAIAH G.S.,
LOUIS A.K.,
NICAISE S.,
SCHEMPP W.,
GROSJEAN C.C. ,
VAN BEEK P.,
On the Laguerre series distribution.
Laguerre and computerized tomography : consistency
conditions and stability of the Radon transform.
Some results on spectral theory over networks,
applied to nerve impulse transmission.
Radar/Sonar detection and Laguerre functions.
Note on two identities mentionned by Professor
Dr. W. Schempp near the end of the presentation of
his paper.
The equation of motion of an expanding sphere in
potential flow.
514
524
532
542
553
555
III. PROBLEHES. COMMENTAIRES PAR A.P. MAGNUS.
1. ASKEY R.,
2. BACRY H.,
3. CALOGERO F.,
4. DEVORE R.A.,
GROSJEAN C.C.,
5. GILEWICZ J.,
6. HAYDOCK R.,
7. KATO Y.,
Two conjectures about Jacobi Polynomials.
An application of Laguerre's emanant to generalized
Chebychev polynomials.
Determinantal representations of polynomials satis-
fying recurrence relations.
Inequalities for zeros of Legendre polynomials.
Solution.
Extremal inequalities for Pade approximants errors
in the Stieltjes case.
Orthogonal polynomials associated to remarkable opera-
tors of mathematical physics; the Hydrogen atom Hamil-
tonian.
About periodic Jacobi continued fractions.
563
564
568
570
571
571
572
574
8. LUBINSKY 0.5.,
9. MAGNUS A.P.,
10. MAGNUS A.P.,
11. MOUSSA P.,
12. MOUSSA P.,
13. NEVAI P.,
14. NEX C.M.M.,
15. van ISEGHEM J.,
16. WIMP J.,
XIII
Diophantine approximation of real numbers by zeroes
of orthogonal polynomials.
Orthogonal polynomials satisfying differential
and functional equations. (Laguerre-Hahn ortho-
gonal polynomials).
Anderson localisation.
Tr(exp(A-XB)) as a Laplace transform.
Diophantine moment problem.
Bounds for polynomials orthogonal on infinite
intervals.
General asymptotic behaviour of the coefficients of
the three-term recurrence relation for a weight func-
tion defined on several intervals.
A lower bound for Laguerre polynomials.
Asymptotics for a linear difference equation.
576
576
577
579
581
582
583
5'S4
5&'4
COfotUNICATIONS NON PUBlIEES DANS CE VOUI£.
BACRY H.,
BARNETT 5.,
BARRUCAND P.,
CALOGERO F.,
An application of Laguerre's emanant to generalized
Chebychev polynomials.
A matrix method for algebraic operations on genera-
lized polynomials.
Problemes lies 8 des fonctions de poids.
Determinantal representations of polynomials satis-
fying linear ode's or linear recurrence relations.
(8 paraitre dans Rend.Sem.Mat.Univ.Politec. Torino 1985)
CASTRIGIANO D.P.L., Orthogonal polynomials and rigged Hilbert space
(8 paraitre dans Journal of Functional Analysis).
DELLA DORA J., RAMIS J.P., THOMANN J., Une equation differentielle
lineaire "sauvage".
DITZIAN Z., On derivatives of linear trigonometric polynomial
approximation process.
DUNKL C.F.,
GREINER P.,
HENDRIKSEN E.,
KATO Y.,
MOUSSA P.,
XIV
Orthogonal polynomials related to the Hilbert
transform. (cfr. Report PM - 88406 C.W.I. Amster-
dam 1984)
The Laguerre calculus on the Heisenberg group.
(cfr. Special functions : Group Theoretical Aspects
and Applications, Ed. R.A. ASKEY, T.H. KOORNWINDER
and W. SCHEMPP. D. Reidel Publishing Company 1984)
A Bessel orthogonal polynomial system.
Proc. Kon. Acad. v. Wet., Amsterdam, ser A,
(1984), 407 - 414.
Periodic Jacobi continued fractions.
Iteration des polyn6mes et proprietes d'orthogonalite.
VAN EIJNDHOVEN S.J.L., Distribution spaces based on classical poly-
nomials.
LISTE DES PARTICIPANTSALFARO M.Departamento de Teoria de FuncionesUniversidad de ZaragozaEspana
ALFARO M.P.Av. de las Torres 93-9°Zaragoza 7Espana
ASKEY R.Department of MathematicsUniversity of Wisconsin480 lincoln DriveMadison, Wisconsin 53706U.S.A.
BACRY H.Centre de Physique TheoriqueLuminy - Case 90713288 MARSEILLE CedexFrance
BARNETT S.School of Mathematical SciencesUniversity of BradfordWest Yorkshire BD7 1DPEngland
BARRUCAND P.151 rue du des Rentiers75013 PARIS
BAVINCK H.Technical UniversityJulianalaan 132DelftNederland
BECKER H.Isarweg 248012 Ottobrunn/MunchenD.B.R.
BElMEHDI S.Univ. Pierre et Marie CurieU.E.R. Analyse, probabilites et Applications4 Place Jussieu75230 Paris Cedex France
BERGERON F.Dept. de Math. et Info.Universite du Quebec a MontrealCase postale 8888, succ. "A"Montreal, P.O. H3C 3P8Canada
BESSIS G. et N.Universite de Lyon ILab. de Spectroscopie Theorique69622 VilleurbanneFrance
BlACHER R.TIM 3 Institut IMAGBP 68Bureau 35, tour I.R.M.A.38402 Saint Martin d'HeresFrance
BREZINSKI cr.Universite de Lille 1U.E.R. I.E.E.A. Informatique59655 Villeneuve d'Ascq CedexFrance
COATMELEC C.8 Rue du Verger35510 Cesson-SevigneFrance
CALOGERO F.Dipartimento di FisicaUniversita di Roma "La Sapienza"Via Sant'Alberto Magno 100153 RomaHalia
CASASUS L.Universidad de la LagunaCatedral, 8 La LagunaTenerifeEspana
CASTRIGIANO D.P.L.Institut fur Mathematik derTechnischen Universitat MunchenArcisstrasse 218000 Munchen 2D.B.R.
COLOMBO S.Rue d'Aquitaine 892160 AntonyFrance
DE BRUIN M.G.Department of MathematicsUniversity of AmsterdamRoetersstraat 151018 WB AmsterdamNederland
DE GRAAF J.Eindhoven University of TechnologyP.O. Box 513EindhovenNederland
DELGOVECentre de Recherche BullLes Clayes Sous Bois78340 France
DELLA DORA J.IMAGUniversite de GrenobleBP 53X38041 Grenoble CedexFrance
DERIENNIC H.H.INSA20, Avenue des Buttes de Coesmes35043 Rennes CedexFrance
DESAINTE-CATHERINE H.Universite de BordeauxUER de Mathematique et Informatique351, Cours de la liberation33405 Talence CedexFrance
DESPLANQUES P.rue Victor Hugo 3959262 Sainghin en MelantoisFrance
DEVILLE H.Unite MEMAUniversite Catholique de louvain1348 louvain-la-NeuveBelgique
DIEUDONNE J.Rue du General Camou 1075007 Paris France
DITZIAN Z.Department of MathematicsUniversity of AlbertaEdmonton T6G 2G1Canada
DRAUX A.Universite de lille 1U.E.R. I.E.E.A. Informatique59655 Villeneuve d'Ascq CedexFrance
DUNKL C.F.Department of MathematicsUniversity of VirginiaCharlottesville - Virginia 22903U.S.A.
DURAND L.University of Wisconsin - MadisonPhysics Dept.1150 University AveMadison - WI 53706U.S.A.
XVI
DUVAL A.3 Rue Stimmer67000 StrasbourgFrance
DZOUHBA J.Univ. Pierre et Marie CurieU.E.R. Analyse,Probabilites et Appl.4 Place Jussieu75230 Paris CedexFrance
GARCIA-LAZARO P.Departamento de MatematicasE.T.S. de IngenierosUniversidad PolitecnicaJose Gutierrez Abascal 2Madrid 6Espana
GASPARD J.P.Universite de LiegeInstitut de Physique - B54000 Sart-Tilman/ Liege 1Belgique
GAUTSCHI W.Purdue UniversityDepartment of Computer ScienceWest Lafayette, IN 47907U.S.A.
GILEWICZ J.CNRS - LuminyCase 907Centre de Physique Theorique13288 Marseille Cedex 9France
GODOY-HALVAR E.Universidad de Santiago de Compostellac/Boan nOl-2Vigo-PontevedraEspana
GREINER P.Mathematics DepartmentUniversity of TorontoToronto Ontario M5S 1A1Canada
GROSJEAN C.C.Seminarie voor Wiskundige NatuurkundeRijksuniversiteit GentGebouw S9Krijgslaan 2819000 GentBelgique
GUADALUPE J.J.Colegio Universitario de La RiojaLogronoEspana
GUADALUPE R.Facultad de QuimicaCastrillo de Aza nO 7-7°AMadrid 31Espana
HAHN W.Alber 88010 GrazAustria
HEN>RIKSEN E.Department of MathematicsUniversity of AmsterdamRoetersstraat 151018 WB AmsterdamNederland
ISERLES A.King's CollegeUniversity of CambridgeCambridge CB2 1STEngland
JACOB G.121, Avenue du Maine75014 PARIS CedexFrance
KANO T.Department of MathematicsFaculty of ScienceOkayama UniversityOkayama 700Japan
KERKER H.Universite de Paris VIIUER de PhysiqueTour 33-432 Place Jussieu75005 Paris
KATO Y.Department of Engineering MathematicsFaculty of EngineeringNagoya UniversityChikusa-kuNagoya 464Japan
KIBLER H.Institut de Physique NucleaireUniversite de Lyon I43 bd du 11 Nov. 191169622 Villeurbanne CedexFrance
XVII
KOORNWINDER I.H.MathematischP.O. Box 40791009 AB AmsterdamNederland
KOWALSKI M.Institute of InformaticsUniversity of WarsawPKIN VIII p. 85000901 WarsawPoland
LAFORGIA A.Dept. di Matematica dell'
Via Carlo Alberto 10TorinoItaly
LAW A.G.University of ReginaSaskatchewan S4S OA2Canada
LEOPOLD E.Centre de Recherche BullLes Clayes Sous Bois78340 France
LOPEZ G.Dept: T. de FuncionesUniversity of HavanaSan Lazaro y L.La HabanaCuba
LOUIS A.K. ..Fachbereich Mathematik, Univ0rsitatErwin-Schrodinger-Strasse6750 KaiserslauternD.B.R.
LUBINSKY 0.5.National Research Institute forMathematical SciencesC.S.I.R.P.O. Box 395Pretoria 0001Republic of South Africa
MAGNUS A.Institut de MathematiqueU.C.L.Chemin du Cyclotron 21348 Louvain-Ia-NeuveBelgique
HARCELLAN F.Departamento de MatematicasE.T.S. de Ingenieros IndustrialesJose Gutierrez Abascal 2Madrid 6Espana
HARONI P.Univ. Pierre et Marie CurieU.E.R. Analyse, Probabilites et Appl.4 Place Jussieu75230 Paris CedexFrance
HASON J.C.Mathematics BranchRoyal Military College of ScienceShrivenhamSwindon, Wilts SN6 8lAEngland
HcCABE J.The mathematical InstituteUniversity of St AndrewsFifeUnited Kingdom
MEIJER H.G.Department of mathematicsUniversity of TechnologyJulianalaan 132DelftNederland
MONTANER-LAVEDAN J.Departamento Teoria de FuncionesUniversidad de ZaragozaEspana
MORAL L.Departamento de MatematicasE.T.S. de Ingenieros IndustrialesUniversidad PolitecnicaJose Gutierrez Abascal 2Madrid 6Espana
HOUSSA P.Service de Physique TheoriqueCentre d'Etudes Nucleaires de Saclay91191 Gif -sur Yvette CedexFrance
HUND E.Service de Metrologie NucleaireU.l.B.Av. F.D. Roosevelt1050 BruxellesBelgique
XVIII
NEVAI P.Department of MathematicsThe Ohio State UniversityColumbus, OH 43210U.S.A.
NEX C.H.H.Univ. of Cambridge - T.C.M. groupCavendisch lab.Madingley RoadCambridge CB3 OH2England
NICAISE S.Universite de l'Etat a MonsDepartement de MathematiqueAv. Maistriau7000 MonsBelgique
OULEOCH£IKH HADJIDU.S. T. Lille I59650 Villeneuve d'Ascq CedexFrance
PASZKOWSKI S.Instytut Niskich Temperatur i BadanStrukturalnych PANPI. Katedralny 150-950 WhJclawPoland
PEREZ GRASA J.Miguel Servet 12 - 80 BZaragozaEspana
PREVOST H.16 Rue de la liberation62930 WimereuxFrance
RAMIREZ GONZALEZ V.Dpto de Ecuaciones FuncionalesFacultad de CienciasAvda Fuente Nueva18001 GranadaEspana
RICHARD F.25 Place des HaIles67000 StrasbourgFrance
RONVEAUX A.Departement de PhysiqueFacultes Univ. N.D. de la Paix61 rue de Bruxelles5000 NamurBelgique
RUNCK£l H.J.Abteilung Mathematik IVUniversitat UlmOberer Eselsberg7900 UlmD.B.R.
SABlONNIERE P.UER IEEA Informatique59655 Villeneuve d'Ascq CedexFrance
SANSIGRE G.Departamento MatematicasE.T.S.I.Jose Gutierrez Abascal 2Madrid 6Espana
SCHEMPP W.Lehrstuhl fur Mathematik IUniversitat SiegenHolderlinstrasse 35900 SiegenD.B.R.
SCHLICHTING G.Math. Inst. Technische UniversitatArcisstrasse 21Postfach 20.24.208000 MunchenD.B.R.
SHAMIR T.Department of Mathematics andComputer ScienceBen Gurion UniversityP.O. Box 653Beer Sheva 84105Israel
STREHl V.Universitat Erlangen-NurnbergInformatik IMartensstrasse 38520 ErlangenD.B.R.
TEf0t4E N.M.Centre for Mathematics and ComputerScienceKruislaan 4131098 SJ AmsterdamNederland
THlJ4ANN J.CNRS Centre de CalculBP 20/Cr67037 Strasbourg CedexFrance
XIX
UllMAN J.l.University of MichiganAnn ArborMichigan 48109U.S.A
VAN BEEK P.Delft University of TechnologyDept. of MathematicsJulianalaan 1322628 BL DelftNederland
VAN EIJNDHOVEN S.Eindhoven University of TechnologyP.O. Box 513EindhovenNederland
VAN ISEGHEM J.9 Allee du Trianon59650 Villeneuve d'AscqFrance
VAN ROSSUH H.Department of MathematicsUniversity of AmsterdamRoetersstraat 151018 WB AmsterdamNederland
VIANO G.A.Dipartimento di Fisica dell'Universita di Genovavia Dodecaneso 3336146 GenovaItalia
VIENNOT G.Universite de Bordeaux IUER de Mathematique et Informatique351 Cours de la Liberation33405 Talence CedexFrance
VINUESA J.Facultad de CienciasApartado 1.021SantanderEspana
VOUE H.Departement de PhysiqueFacultes Univ. N.D. de la Paix61 Rue de Bruxelles5000 NamurBelgique
WIMP J.Drexel UniversityPhiladelphia Pa 19104U.S.A.
WUYTACK L.Department of MathematicsUniversity of AntwerpUniversiteitsplein 1B 2610 WilrijkBelgium
lOllA F.22 rue Montpensier64000 PauFrance
xx
EDMOND NICOLAS LAGUERRE
Claude Brezinski
Universite de Lille I
59655 - Villeneuve d'Ascq cedex
France
Edmond Nicolas Laguerre naquit rue Rousseau, a Bar-Le-Duc dans le
departement de la Meuse, le 9 avril 1834 a une heure du matin. Il
etait le fils de Jacques Nicolas Laguerre, marchand quincallier, age
de trente sept ans et de son epouse Christine Werly.
Il fit ses etudes dans divers etablissements pUblics, ses parents
l'ayant successivement place au college Stanislas, au lycee de Metz
et a l'institution Barbet afin qu'il eut toujours aupres de lui un
camarade pour veiller sur sa sante deja precaire. Il montrait une rare
intelligence avec un gout prononce pour les langues et les mathema-
tiques. Ses premiers travaux sur l'emploi des imaginaires en geometrie
remontent aux annees 1851 et 1852 et son premier article parut en 1853
dans les Nouvelles Annales de Mathematiques dirigees par Terquem qui
note alors : "Profond investigateur en geometrie et en analyse, le
jeune Laguerre possede un esprit d'abstraction excessivement rare, et
l'on ne saurait trop encourager les travaux de cet homme d'avenir".
Il donnait la solution complete du probleme de la transformation homo-
graphique des relations angulaires, completant et ameliorant ainsi les
travaux de Poncelet et Chasles.
Le ler novembre 1853 il entre quatrieme sur cent-dix a l'Ecole
Polytechnique. D'apres son signalement il mesure 1,685 m., ales
cheveux et les sourcils chatain clair, le front haut, le nez moyen,
les yeux gris bleus, la bouche large, le menton rond, le visage long.
Il est myope et a un signe pres de l'oreille gauche. Ses professeurs
sont J.M.C. Duhamel et C. Sturm pour l'analyse et de La Gournerie pour
la geometrie.
Pendant l'annee scolaire 1853-1854, on il occupe l'emploi de
sergent-fourrier, ses professeurs font les observations suivantes sur
son travail :
XXII
"Travail assidu mais qui pourrait !tre mieux regIe."
Notes d'interrogations particulieres : constamment bonnes ou tres
bonnes en analyse; d'abord tres bonnes mais constamment decrois
santes depuis Ie commencement du semestre en geometrie
trop variables en physique ; tres bonnes en chimie.
Notes d'interrogations generales: mediocre en analyse tres
bonne en geometrie descriptive."
Pour Ie second semestre on trouve
"Resultats bons ou assez bons dans toutes les parties, mais moins
satisfaisants en general que ceux du premier semestre".. ieme . iemeEn effet 11 est 11 au classement du prem1er semestre et 24 au
second.
Quant a sa conduite les appreciations sont moins favorables :
"Conduite assez bonne. Tenue mauvaise. Eleve leger et bruyant".
II plusieurs punitions pour mauvaise tenue, bavardage et chant
pendant l'etude.
II passe en seconde annee 59 i eme sur 106. En 18541855, on Ie
juge ainsi :
"Travail soutenu. Notes generalement bonnes ou tres bonnes en
analyse, en mecanique et en physique; tres mediocres en chimie."
La conduite et la tenue sont passables. Par contre il est toujours
"tres causeur et tres negligent" et evidemment il "aurait pu beaucoup
mieux faire". II est puni de deux jours de salle de police pour avoir
"allume du feu dans l'etude".
II sort de l'Ecole Polytechnique 46 i eme sur 94 avec les apprecia
tions suivantes :
"Cet eleve tres intelligent aurait pu rester classe dans les pre
miersde sa promotion, mais n'a pas travaille. Extr !ment dissipe.
Doit et peut tres bien se poser a l'Ecole d'application."
Son classement de sortie lui ferme l'acces aux carrieres civiles.
II entre 7 i eme sur 41 a l'Ecole Imperiale d'Application de l'Artillerie
et du Genie a Hetz, Ie ler mai 1855. II ne semble pas !tre plus atten
tif qu'a Polytechnique
"Conduite bonne mais a souvent ete puni pour retards dans ses
travaux. Tenue bonne, mais tournure peu militaire. A des moyens
pour les mathematiques, mais n'a aucun gout pour les travaux gra
phiques, dessine mal et lentement. S'est trop occupe d'objets
etrangers aux etudes de l'ecole. C'est l'officier qui a Ie plus
de retard dans ses travaux. Parle un peu l'Italien".
XXIII
II sort de l'ecole 32 i eme sur 40 et Ie general inspecteur note
"A perdu beaucoup de rangs parce que, sans Atre paresseux, il
s'est occupe de choses etrangeres aux travaux de l'ecole. C'est
un travers dont il pourra se corriger."
A sa sortie de l'Ecole de l'Artillerie il entame une carriere
militaire. Il est SOUS lieutenant au 3eme regiment d'artillerie a pied
le 6 decembre 1856 puis lieutenant 1e l e r mai 1857. Le 13 mars 1863
il est nomme capitaine et est employe, comme adjoint, a la manufacture
d'armes de Mutzig. Le 18 juin 1864 il abandonne cet emploi pour deve
nir repetiteur adjoint au cours de geometrie descriptive a l'Ecole
Polytechnique.
Le 17 aout 1869 il epouse r1arie Hermine Albrecht, fille de Julie
Caroline Durant de r1areuil, veuve de Leopold Just Albrecht, decede,
proprietaire, demeurant au chateau d'Ay dans Ie departement de la
r1arne. Sa femme re90it en dot 24000 francs en actions nominatives
produisant 1200 francs de revenus. De ce mariage naitront deux filles.
A cette epoque il habite 3 rue Corneille a Paris, plus tard il habi
tera 61 boulevard Saint Michel.
En novembre 1869 il est autorise a faire un cours de geometrie
superieure a la Sorbonne.
Pendant Ie siege de 1870 il est d'abord designe, Ie 28 aout, par
Ie General Riffault pour commander en second la batterie de rempart,
dite de l'Ecole Polytechnique. Le 12 novembre il est nomme au comman
dement de la 13i eme batterie du regiment d'artillerie et prend part,
en cette qualite, aux deux combats de Champigny le 30 novembre et Ie
2 decembre 1870. Pour sa conduite, il est fait chevalier de la Legion
d'honneur le 8 decembre.
Pendant l'insurrection de Paris il "a conserve jusqu'au 27 mars
le commandement des hommes qui restaient dans la batterie, licenciee
en partie Ie 14 mars. Apres dissolution forcee de la batterie, a re
joint a Tours l'Ecole Polytechnique au il avait ete reclasse".
Apres ces evenements il reprit ses enseignements a Polytechnique
ainsi que ses travaux scientifiques. Le 25 navembre 1873 il est norome
repetiteur du cours d'analyse a polytechnique et examinateur d'admis
sion le 4 mai 1874, charges qu'il canservera jusqu'a sa mort. Le 31
mai 1877 il passe au grade de Chef d'escadron. II est "tres aime et
tres estime" a l'Ecale Polytechnique. En 1880 l'inspecteur general
XXIV
note dans son dossier
"Excellent repetiteur d'analyse, le Cowmandant Laguerre occupe un rang
distingue parmi nos jeunes geometres et il a devant lui un bel avenir
de savant". 11 avait deja publie alors 114 articles
Le 5 juillet 1882 il est fait officier de la Legion d'honneur.
Afin de pouvoir se consacrer entierement a ses travaux, il prend une
retraite anticipee le 2 juin 1883.
Le 11 mai 1885 il est elu a l'Academie des Sciences grace a l'ac
tion de Camille Jordan qu'il avait connu quand ils etaient tous les
deux eleves de Polytechnique. Peu de temps apres Joseph Bertrand lui
confiait la suppleance de la Chaire de Physique Mathematique au College
de France. 11 y fait un cours tres remarque sur l'attraction des ellip
soides.
Sa sante deja faible et une fievre continuelle le contraignirent
a abandonner toutes ses occupations. 11 revint a BarLeDuc a la fin
de f ev.r i.e.r 1886. Laguerre mourut le 14 aotrt; 1886a4 heuresdumatin au
52 rue de Tribel. Georges Henri Halphen representa l'Academie a ses
obseques et quelques mots apres avoir lu un discours de
Joseph Bertrand.
Sources documentaires :
Archives de l'Ecole Polytechnique.
Archives du Service Historique de l'Armee de Terre.
E.N. Laguerre: Notice sur les travaux mathematiques, Gauthier
Villars, Paris, 1884.
E. Rouche : Edmond Laguerre, sa vie et ses travaux, J. Ec.
Polytech., Cahier 56 (1886) 213271.
C.R. Acad. Sci. Paris, 103 (1886) 407.
Nouv. Ann. Math., (3) 5 (1886) 494496.
C.R. Acad. Sci. Paris, 103 (1886) 424425.
H. Poincare : Notice sur la vie et les travaux de M. Laguerre,
membre de la section de geometrie, C.R. Acad. Sci. Paris, 104
(1887) 16431650.
A. de Lapparent : Laguerre, Livre du Centenaire de l'Ecole Poly
technique, GauthierVillars, Paris, 1895, tome 1, pp. 149153.
L'Ecole Polytechnique, GauthierVillars, Paris, 1932, pp.
M. Bernkoff : Laguerre, Dictionary of Scientific Biography, C.C.
Gillispie ed., C. Scribner's sons, NewYork, 1973.
E.N. Laguerre: Oeuvres, reprint by Chelsea, NewYork,1972,2vols.
xxv
XXVI
LAGUERRE
AHD
ORTHOGOHAL POLYNOnIALS :IN 1984 •
by A.P. Magnus and A. Ronveaux
The importance of orthogonal polynomials can be estimated from the
following statistics
Up to 1940 , one finds about 2000 entries in the Shohat , Hille and
Walsh bibliography [4] •
C.Brezinski's bibliography [2] on orthogonal polynomials and the
related subjects of Pade approximation and continued fractions ,
contains now more than 5000 titles •
The MATHF:ILE data base allows variously tuned quests : since 1973 ,
one finds 2984 titles and abstracts containing the words 'orthogonal'
AHD 'polynomial(s), ,
families :
but one must also add references to special
Cebysev polynomial(s) 1193
Hermite polynomial(s) 1290
Jacobi polynomial(s) 1283
Laguerre polynomial(s) 1167
Legendre polynomial(s) 1124
Bessel polynomial(s) 224
The other special orthogonal polynomials (Charlier , Hahn ,
Krawtchouk , Meixner) have a much smaller record (from 20 to 50) •
The name of Laguerre appears 1405 times, showing that his present
influence is mainly centered on polynomials (each of the other names
XXVIII
is in more than 2000 titles and abstracts • excepting Bessel : 1264) .
This is emphasized by Bernkopf [1] who mentions only briefly
Laguerre's achievements in geometry (once famous). but gives a
detailed account of the paper introducing what are now called Laguerre
polynomials (Sur l' integrale [
- xe dX.
x xBull. Soc. l1ath. France
1(1879) =[3] vol.1. pp.428-438) R.Askey ([5]. vol.3 p.866) •
looking for the various appearences of the Laguerre polynomials before
Laguerre • finds two papers of R. l1urphy (Trans. Camb. Phil. Soc. !!(1833)355-408. 2(1835) 113-148) as their birthplace. We conclude
that the Laguerre polynomials are about as old as Laguerre himself
(150 years) .
To be honest. one must remark that Laguerre used his virtuosity
in geometry when dealing with polynomials. especially with the
location of their zeros. These works ([3]. vol.1) are still
influential. and so are the author's methods: just consider the
title of the famous book by 11.l1arden: 'Geometry of Polynomials'
(AI1S • Providence ,2nd ed. 1966); see also Bacry's contribution in
the present volume .
To return to orthogonal polynomials in Laguerre's output • a number
of papers written in the period 1877-1885 ([3]vol.1. 318-335,
438-448, vol.2, 685-711), the last one (= J.de l1ath. 1 (1885)
135-165) being the most important. explore the properties of
orthogonal polynomials related to weight functions satisfying
p'(x)/p(x) = a rational function of x •
(up to a finite number of Dirac S functions). Actually. Laguerre
studied Pade approximations and continued fraction expansions of
functions satisfying a differential equation of the form
W(z)f'(z) = 2V(z)f(z) + U(z)
XXIX
whe:re W, V and U a:re polynomials [see I'IcCabe's cont:ribution] • One
:recove:rs [possibly fo:rmal] o:rthogonal polynomials as denominato:rs of
app:roximants of f if fez) can be r.r:ritten as a definite integ:ral
fs(Z-X) - 1p(X)dX , with p positive on a :real set S , the denominato:r Pn
of the [n/n] Pade app:roximant of f is the nth deg:ree o:rthogonal
polynomial :related to p ; if such an integ:ral fo:rm does not hold , butQ)
if f has an expansion fez) = L cnz-n-1, Pn is called a fo:rmaln=O
o:rthogonal polynomial. In the fi:rst case, the rational function
p'(x)/p(x) is p:recisely 2V(x)/W(x) the connection has been made
clea:r by Shohat ['Su:r une classe etendue de f:ractions continues
algeb:riques et su:r les polynomes de Tchebycheff cor:respondants' , C.R.
Acad. Sci. Pa:ris ill( 1930) 989-990; 'A diffe:rential equation fo:r
o:rthogonal polynomials' , Duke l'Iath.J. (1939) 401-417 ]
Lague:r:re succeeded in showing that the o:rthogonal polynomials Pn
satisfy :rema:rkable diffe:rential equations-
W8ny" + [(2V+W')8n-W8'n]y' + Kny = 0 ,
whe:re 8n and Kn a:re polynomials, whose coefficients a:re solutions of
ce:r:tain (usually) nonlinea:r equations. The deg:r:ees of 8n and Kn aze
bounded by and , whe:r:e = max«deg:ree V) -1, (deg:ree W) -2 ) •
The equations involve an inte:rmediate set of polynomials {Qn} of
deg:ree , and a:re
(x-sn)(Qn+1(X)-Qn(x» + 8n+1(x) - :rn8n_1(x)/rn_1 = W(X)n=O,1, •..
Qn+1(X) + Qn(x) = -(x-sn)8n(x)/:rnwith 80=U , Qo=V , 8_1/:r:_1=0. The :rn's and sn's a:re the coefficients
- For a sophisticated algebraic geomet:ry p:resentation, see 'Pade
app:roximation and the Riemann monod:romy p:roblem', by G.V.
Chudnovsky, pp449-510 in 'Bifu:rcation Phenomena in l'Iathematical
Physics and Related Topics', edited by C.Ba:rdos and D.Bessis,
D.Reidel , Do:rd:recht 1980 .
xxx
usefulvezy
= (x-sn)Pn(x)
the polynomials
then given by
azeQn'sThe
of the th%ee-term zecurzence zelation Pn+1(X)
-ZnPn-1(X). and aze found when one expzesses that
9n 's keep a degzee SM. The polynomials Kn aze
n-1+9 n L 9k / Z k •
k=O
themselves as they entez diffezential zelations Wp'n = (Qn-V)Pn +
9nPn-1 (this is the basis of quasi-ozthogonality chazactezizations
tzeated zecently by Bonan • Hendziksen • Lubinsky. Mazoni. Nevai.
Ronveaux • van Rossum) •
This vezy elabozated wozk has been zightly called a mastezpiece by
R.Askey in his talk during the meeting Neaz the end of his
contzibution with G.!. Andzews you will find a challenge apply
Laguezze's theozy to theiz wide extended set of classical ozthogonal
polynomials .•• Actually. the concept of diffezential equation must
also be extended to diffezence oz functional equation. The zequized
matezial is to be found in W.Hahn's most impzessive contzibution •
togethez with faz-zeaching invezse theozems .
The "classical" classical ozthogonal polynomials aze zecovezed by
solving Laguezze's equations in the simplest case M=0 (degzees of W
and V bounded by 2 and 1 ). This is explained in Hendziksen and van
ROssum's contzibution in the pzesent volume (see also theiz papez 'A
Pade type appzoach to non-classical ozthogonal polynomials'. in
J.Math.An.Appl. 106. 237-248' (1985). wheze Bessel polynomials aze
also considezed) . The Laguezze equations aze then exactly
solvable. as shown by Laguezze himself foz the exemples of the
Legendze and ••• the Laguezze polynomials (even the extended ones) •
When M>0 • a genezal way to solve the equations is still not known
but special cases have been tzeated. often by people unawaze of
Laguezze's work. as the Kzall's. Littlejohn. Koornwindez... [seeCIC ,.
'Orthogonal polynomials with weight function (1-x) (1+x) + +
Canad. Math. Bull. :1(2). 205-214 (1984). by the last
XXXI
author ] as rema:r:ked by Hendriksen and van Rossum in their quoted
paper. Freud, Bonan and Nevai also rediscovered some instances of
Laguerre' equations when Wis a constant, but V of a:r:bitra:r:y degree ,
so that p is the exponential of a polynomial (see A.P.Magnus'
contribution) .
In the last pages of his paper of 1885 ([3] vol.2, 685-711),
Laguerre began the study of the case M =1 (degrees of Wand V 3 and
2 • equivalent to W(X) , V(x)/x , U(x) even of degrees 4 , 2 , 2 ) .
He recognized the importance of elliptic integrals and Abelian
functions in the solution of this problem , but was stopped by illness
and death Establishing asymptotic estimates is already terribly
difficult Gammel and Nuttall ('Note on generalized Jacobi
polynomials' , pp.258-270 in Lect. Notes Math. 925) predicted indeed
that, if the three zeros b1 , b2 , b3 of Wa:r:e distinct and not
collinea:r:, the asymptotic behaviour of Pn(x) and related functions
involves elliptic integrals of the form J:(t-a)1'2(W(t»-1'2dt , where
a and e a:r:e constants (a is the center of capacity of b1 , b2 and
b3 ). The asymptotic form was deduced from the Liouville-Green
approximation to the solution of the Laguerre differential equation .
Some assumptions had to be made, because en' a factor of y" in the
differential equation, is now of degree 1 and vanishes therefore at
some point Zn. However, it happens that no solution of the
differential equation is singula:r: at this point Zn is an appa:r:ent
singula:r:ity . Such appa:r:ent singula:r:ities a:r:e unavoidable when dealing
with non elementa:r:y cases (Hahn). In order to settle asymptotic
behaviour, it is important to control the wanderings of Zft. The
central expression in Liouville-Green's estimates is
Jx[ Kn(t) ] 1'2C W(t)(t-z
n)dt and it was assumed that the two zeros of Kn a:r:e
close to a and to Zn, in order to get the desired expression. A
XXXII
complete of the asymptotics. avoiding
assumptions. has now been by J.Nuttall ('Asymptotics of
Jacobi polynomials' • submitted to ) • who
constEucts the OlveE's pathes of
integEation, using of H.Stahl ('The of Pade
app:r:oximants to functions with b:r:anch-points'. p:r:ep:r:int). This
settles only the case IA =1. but the same ideas aze expected to be
valuable in gene:r:al (see 'Asymptotics of diagonal He:r:mite-Pade
polynomials' • J.AppEOX. TheoEY , 42 (1984) • 299-386 by J.Nuttall fo:r:
the whole ) .
[1] M.BERHKOPF • LagueE:r:e • Edmond Nicolas • DictionaEY of Scientific
BiogEaphy pp.573-576, C.C.GILLISPIE edito:r:. ChaEles Sc:r:ibne:r:'s
Sons • New Yo:r:k 1973 .
[2] C.BREZINSKI, A BibliogEaphy on Pade App:r:oximation and Related
Subjects . Publications de Lille I • 1977-1982 •
[3] E.N.LAGUERRE. OeuvEes. Ch.HERMITE. H.POINCARE. E.ROUCHE
• 2 vol. • PaEis 1898 &1905 • =Chelsea • New Yo:r:k 1972 .
[4] J.A.SHOHAT, E.HILLE. J.L.WALSH, A on OEthogonal
Polynomials. Bull. Nat.Res. Council nO 103 , Washington 1940 •
[5] G.SZEGO, Collected Paper:s, R.ASKEY editor:, 3 vol. ,
, Boston , 1982 .
XXXIII
With 60 cont:r:ibutions, (mo:r:e than 75 when one includes
the p:r:oblems , :r:ep:r:esenting t:r:ends of futu:r:e :r:esear:ch ) , one may hope
that almost all the living aspects of the subject ar:e cove:r:ed in this
book. A gene:r:al su:r:vey can be found in the invited cont:r:ibution of
J .Dieudonn'. One will app:r:eciate that many autho:r:s of var:ious
sections we:r:e inspi:r:ed by some of Lague:r::r:e's own wo:r:ks .
Section 1, concepts of o:r:thogonality, contains wo:r:ks desc:r:ibing
the consequences of defining o:r:thogonality by specific functionals •
These studies on fo:r:mal o:r:thogonality ar:e :r:elated to Pad'
app:r:oximation and its nume:r:ous applications (app:r:oximation , nume:r:ical
analysis , •.. ). The production of recu:r:rence relations is usually a
major :r:equirement in these questions, but one may also start with
such relations (see the invited pape:r: by W.Hahn) •
Combinatorics and graph theory ar:e related to orthogonal
polynomials in a way that will perhaps be a discovery for some readers
of our second section Unexpected connections and ingenious
derivations ar:e present , but also a way towar:ds various applications.
No wonder that similar: tools appear: in some othe:r: contributions
solid-state phYsics (J.P.Gaspard &Ph.Lambin) , netwo:r:ks (S.Hicaise)
The third section is devoted to functional analysis aspects.
Algebra (of operators) and topology (in sequence spaces or
[generalized] functions spaces) meet here, introducing convergence
considerations that will of course reappear in many other sections •
One may recall that the fundamentals of the analysis of orthogonal
polynomials come from spectral properties of tridiagonal operators
(Jacobi matrices) acting on Hilbert spaces (J.Dieudonne ) .
XXXIV
One can define oxthogonal polynomials with xespect to sets of the
complex plane. A vexy active Spanish school pxesents its xeseaxches
in this field in section 4. The contxibutions of the llfaxo's •
G.Lopez and P.Hevai axe also linked to this subject •
Classical. but often difficult mattexs of mathematical analysis
axe connected with the study of measures and the xelated oxthogonal
polynomials. especially as fax as asymptotic pxopexties axe
concexned. See also G.Lopez and A.Magnus in other sections than the
pxesent one (which is the nO 5) Rakhmanov's theoxem. a majox
advance in this field. is commented extended and used in Hevai's
and Lopez' contxibutions •
The pattexns of zexos of oxhogonal polynomials axe impoxtant in
many applications. Most of the contxibutions to this section 6 deal
with accurate (ox shaxp) estimates. Thexe is also an unexpected
xeconstxuction of moments fxom extxeme zexos (invexse pxoblem) •
Anothex phenomenon xelated to zexos is given by H.Stahl in next
section •
The use of oxthogonal polynomials in appxoximation theoxy is
considexed in section 7. This subject is closely xelated to
appxoximation and vaxious genexalizations. Special oxthogonal sexies
axe also considexed elsewhexe. especially in section 9 (numexical
analysis) •
xxxv
Special families of orthogonal polynomials are characterized by a
finite number of parameters Up to now, classical orthogonal
polynomials form a very impressive five parameters family
(G.E.Andrews &R.Askey, where you can also find the information of
Labelle's "Tableau d'Askey" , instead of damaging your eyes) •
The constraints represented by the existence of functional equations
define also special families (W.Hahn) •
This section 8 contains many contributions about special families ,
old or new, classical or not, characterized by their weight
function, recurrence relation, differential properties, etc •.. See
also the two next sections for applications and Koornwinder's
contribution in section 3 .
Special families also help in making progress in apparently unrelated
domains of analysis. The final proof of a very famous conjecture ,
and how some participants to the meeting were involved in it , was the
subject of many admirative comments .•. (of course, we mean here
W.Gautschi, R.Askey and Bieberbach 's conjecture, see 'Et la
conjecture de Bieberbach devint Ie theoreme de Louis de Branges ... ' by
C.A.Berenstein and D.H.Hamilton , La Recherche 1ft (1985) 691-693 ) •
The invited contribution of W.Gautschi and the contents of section
9 deal with the numerical analysis of orthogonal polynomials .
Progresses in constructive stable methods of obtention, ingenious
algorithms, use in approximation and representation of functions ,
work with series are presented here (see also A.Iserles &in section 1 for ODE solvers) .
Applications to the non-mathematical world (but presented in a
fair mathematical way) follow in section 10. One finds study of
matter, models of complex systems, including biological ones,
signal analysis, statistical tools. Investigations on the editors
brains are sadly missing (can be left as a problem) .
TABLEAU D'ASKEY
Jacques Labelle.
Universite du Quebec a MontrealOepartement de Mathematiques et Informatique
Case Postale 8888. Succursale "A"Montreal PO. H3C3P8
CANADA
La figure ci·-contre presente une reduction d 'un tableau resumant les proprie-
tes des poLynomes orthogonaux classiques (au sens de [ n). Les relations entre
ces polynomes sonl: egalement figurees, demontrant la profonde unite de l'ensemble.
Ce tableau tente de realiser un voeu exprLme par R. Askey, qui l' ad' ailleurs rea-
lise LuL-meme dans un ouvrage recent [ 2J .
Les details devenus invisibles (les dimensions originelles sont de 122 cmx89 em).
peuvent etre reconstitues a la lecture du texte d' Andrews et Askey [1J.. On peut
aussi s' adresser l' auteur.
A noter que les q·-analogues n'ont pas ete present.es , leur inclusion necessitant
un graplE a trois dimensions (refLexi.cn commun Lquee pa.r R. Askey).
[1] G.E. ANDREWS, R. ASKEY Classical orthogonal polynomials, dans ce volume.
[2J R. ASKEY,. J .A. WILSON, Some basic hypergeometric orthogonal polynomials
that generalize Jacobi polynomials. Memoirs Amer. Math. Soc. 1985.
Liol.c>
J-I,.J......tI- t ,t'lel i;fr·:-:t.":;:··-.f).'-,.t t...
l(..·e"7"'l.•..a .. •.rtt.ht , .•4."_
•.v-. ttl_'_' .'_._
f:i:: :' :'!- :::'" ;/; ...(f _ •• -oI(j + .•• ..
..I"·...jfl.....-.,j· - ....
'!i'....... .at
-,'(......-t....",...4.._WO'r-,c...1(..1.. •.. ...-,.,.('.,
"--
XXXVII
hll., ,."':l1":;..... ..•).-<# , .. •.,....'=··
(1-,l·(I·.r-..t;Ifft.I·'ff-..c.lI'IJ..t·l·l(..
..•'''"..,·....... 1'I:..:...'":::":":".:::=:..-J.'.I.hll