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

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