Polynôme de Gegenbauer

Cet article est une ébauche concernant les mathématiques.

Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants.

En mathématiques, les polynômes de Gegenbauer ou polynômes ultrasphériques sont une classe de polynômes orthogonaux. Ils sont nommés ainsi en l'honneur de Leopold Gegenbauer (1849-1903). Ils sont obtenus à partir des séries hypergéométriques dans les cas où la série est en fait finie :

C_{n}^{(\alpha )}(z)={\frac {(2\alpha )^{\underline {n}}}{n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right)

où $n$ est la factorielle décroissante^[1].

Propriétés

Orthogonalité

Les polynômes de Gegenbauer sont orthogonaux sur [-1 ; 1] pour le poids $w (x) = (1- x 2) α -1/2$ :

$\int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\mathrm {d} x=\delta _{n,m}{\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )\Gamma (\alpha )^{2}}}$
Récurrence

Les polynômes de Gegenbauer peuvent être construits par la relation de récurrence :

$C_{0}^{(\alpha )}(x)=1,\ C_{1}^{(\alpha )}(x)=2\alpha x,\ C_{n}^{(\alpha )}(x)={\frac {1}{n}}\left(2x(n+\alpha -1)C_{n-1}^{(\alpha )}(x)-(n+2\alpha -2)C_{n-2}^{(\alpha )}(x)\right)$
Liens avec d'autres suites de polynômes orthogonaux

Les polynômes de Gegenbauer sont solutions de l'équation différentielle :

(1-x^{2})y''-(2\alpha +1)xy'+n(n+2\alpha )y=0.\,

On peut alors remarquer que pour $α = 1/2$ , l'équation se ramène à celle satisfaite par les polynômes de Legendre, et pour $α = 1$ , on retrouve celle des polynômes de Tchebychev de seconde espèce.

Polynôme de Gegenbauer $C (1) n (x)$
Polynôme de Gegenbauer $C (2) n (x)$
Polynôme de Gegenbauer $C (3) n (x)$

Applications

Les polynômes de Gegenbauer apparaissent comme des prolongements des polynômes de Legendre dans la théorie du potentiel pour les dimensions supérieures à 1.

Notes et références

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Gegenbauer polynomials » (voir la liste des auteurs).

↑ Voir (en) Milton Abramowitz et Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [détail de l’édition] (lire en ligne), p. 561.

Liens externes

(en) Eric W. Weisstein, « Gegenbauer Polynomial », sur MathWorld

Portail des mathématiques

Read other articles:

University of Virginia

Public university in Charlottesville, Virginia, US UVa redirects here. For other uses, see Uva. University of VirginiaTypePublic research universityEstablishedJanuary 25, 1819; 205 years ago (January 25, 1819)[1]FounderThomas JeffersonAccreditationSACSAcademic affiliationsAAUORAUSCHEVURASea-grantSpace-grantEndowment$13.6 billion (2022)[2]Budget$1.91 billion (2020)[3][a]PresidentJames E. RyanProvostIan BaucomAcademic staff3,265 (Fall 2019)[...

Oscar W. Gillespie

American politician This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: References. Please help improve this article if you can. (December 2022) (Learn how and when to remove this template message) Oscar William GillespieMember of the U.S. House of Representativesfrom Texas's Twelfth districtIn officeMarch 4, 1903 – March 3, 1911Preceded byJames L. SlaydenSucceeded byOscar Callaway Personal detailsBorn(1858-06-20)June 20, 185...

Steve Heighway

Steve Heighway Informasi pribadiNama lengkap Stephen Derek HeighwayTanggal lahir 25 November 1947 (umur 76)Tempat lahir Dublin, Ireland[1]Posisi bermain WingerKarier junior Skelmersdale UnitedKarier senior*Tahun Tim Tampil (Gol)1970–1981 Liverpool 329 (50)1981 Minnesota Kicks 26 (4)1981–1982 Philadelphia Fever (indoor) 23 (3)Total 378 (57)Tim nasional1970–1981 Republik Irlandia 34 (0) * Penampilan dan gol di klub senior hanya dihitung dari liga domestik Stephen Derek Heigh...

Nazik Al-Hariri Welfare Center for Special Education

School in Amman, JordanNazik Al-Hariri Welfare Center for Special EducationAddressQusoor Malakeah RoadDabouqAmmanJordanInformationSchool typeSpecial educationEstablished1985FounderNazik HaririStaff60Enrollment170Campus size13 000 m2 Nazik Al-Hariri Welfare Center for Special Education was founded by Nazik Hariri in 1985 in Amman, Jordan.[1] It provides students with special needs with social care, education, rehabilitation and recreational care by helping them acquire daily life skill...

Перуанский анчоус

Перуанский анчоус Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:ЧелюстноротыеГруппа:Костные рыбыКласс:Лучепёрые рыбыПодкласс:Новопёрые �...

Harry Willis Miller

American physician and Seventh-day Adventist missionary Harry Willis MillerBornJuly 1, 1879Ludlow Falls, OhioDiedJanuary 1, 1977Riverside, CaliforniaOccupation(s)Physician, Seventh-day Adventist missionary Harry Willis Miller (July 1, 1879 – January 1, 1977) was an American physician, thyroid surgeon and Seventh-day Adventist missionary. Miller was a vegetarian and pioneer in the development of soy milk.[1] Biography Miller was born in Ludlow Falls, Ohio on July 1, 1879.[2] ...

哥伦布 (密西西比州)

密西西比州哥伦布城市綽號：Possum Town哥伦布位于密西西比州的位置坐标：33°30′06″N 88°24′54″W / 33.501666666667°N 88.415°W / 33.501666666667; -88.415国家美國州密西西比州县朗兹县始建于1821年政府 • 市长罗伯特·史密斯 (民主党)面积 • 总计22.3 平方英里（57.8 平方公里） • 陸地21.4 平方英里（55.5 平方公里） • ...

اضطهاد ديني

جزء من سلسلة حولالتمييز أشكال عامة عمر طائفة طبقة لون إعاقة نمط وراثي شعر طول لغة مظهر سمات عقلية عرق / أثنية / جنسية رتبة دين جنس توجه جنسي حجم أنواع أشكال محددة اجتماعية رهاب اللاجنسية وصمة عار الإيدز سلطة البالغين اضطهاد المصابين بالبرص معاداة التشرد معاداة الم�...

Elverum

Municipality in Innlandet, Norway This article is about the municipality in Innlandet, Norway. For other uses, see Elverum (disambiguation). Municipality in Innlandet, NorwayElverum Municipality Elverum kommuneMunicipalityLudgaardsvegen at Elverum square, 2022 FlagCoat of armsInnlandet within NorwayElverum within InnlandetCoordinates: 60°55′39″N 11°42′55″E / 60.92750°N 11.71528°E / 60.92750; 11.71528CountryNorwayCountyInnlandetDistrictØsterdalenEstablished...

Jordan Webb

Jordan Webb Webb playing for Tampines Rovers in 2016Informasi pribadiTanggal lahir 24 Maret 1988 (umur 36)Tinggi 182 m (597 ft 1+1⁄2 in)[1]Posisi bermain WingerInformasi klubKlub saat ini Tampines RoversKarier junior East York KometsKarier senior*Tahun Tim Tampil (Gol)2009 Italia Shooters 2010–2012 Hougang United 77 (33)2013 Home United 24 (8)2014–2016 Young Lions 39 (14)2016 Tampines Rovers 22 (12)2017 Warriors 23 (7)2018– Tampines Rovers 0 (0) * Pena...

Perang Salib Swedia Pertama

Erik IX dari Swedia dan Uskup Henry dalam perjalanan ke Finlandia. Lukisan Abad Pertengahan Akhir dari Uppland. lbsPerang SalibDi Tanah Suci (1095–1291) Pertama Rakyat 1101 Norwegia Venesia Kedua Ketiga 1197 Keempat Kanak-Kanak Kelima Keenam Para Baron Ketujuh Para Gembala 1251 Kedelapan Kesembilan Sesudah 1291 Fakir Miskin 1309 Para Gembala 1320 Smirna 1343–1351 Aleksandria 1365 Savoia 1366 Barbaria 1390 Nikopolis 1396 Varna 1443 Portugal 1481 Perang Salib Utara (1147–1410) Wen...

Komsomolets Island

Island of the Severnaya Zemlya group in the Russian Arctic This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Komsomolets Island – news · newspapers · books · scholar · JSTOR (March 2012) (Learn how and when to remove this message) KomsomoletsNative name: остров КомсомолецTerra-MODIS image of Ko...

Via Appia

Disambiguazione – Se stai cercando la moderna strada statale, vedi Strada statale 7 Via Appia. Via AppiaVia AppiaVia Appia: regina viarumLocalizzazioneStato attuale Italia RegioneLazioCampaniaBasilicataPuglia Coordinate41°50′29″N 12°31′57″E41°50′29″N, 12°31′57″E Informazioni generaliTipostrada romana Costruzione312 a.C.-dopo il 244 a.C. CostruttoreRepubblica romana, Appio Claudio Cieco Condizione attualetratto iniziale a Roma ben conservato. Molti ponti e tratti an...

Aureus

Aureus utgivet år 193 av Septimius Severus Aureus (egentligen aureus denarius) är ett romerskt guldmynt. Guldmynt präglades under den romerska republiken endast i undantagsfall, framför allt av segrande fältherrar som Sulla och Pompejus. Först Julius Caesar införde guldet som regelmässig myntmetall. De guldmynt han slog vägde 1/40 libra (8 gram). Under den efterföljande tiden sjönk guldmyntens vikt, Nero stabiliserade den till 1/45 libra (7,3 gram). Från och med Marcus Aurelius bl...

Joyzelle Joyner

Joyzelle JoynerJoyner dalam film The Sign of the Cross (1932)Lahir27 Agustus 1905Meninggal30 November 1980(1980-11-30) (umur 75)Orange, California, Amerika SerikatPekerjaanPemeranTahun aktif1925–1935 Joyzelle Joyner (27 Agustus 1905 – 30 November 1980) adalah seorang pemeran dan penari Amerika Serikat. Ia tampil dalam sekitar tiga puluh film antara 1925 dan 1935, serta meraih ketenarannya karena tampil dalam The Sign of the Cross. Referensi Pranala luar Wikimedia ...

Green Lane Hospital, Auckland

Public hospital in Auckland, New Zealand For the hospital in Devizes, England, see Green Lane Hospital. Hospital in Auckland, New ZealandGreen Lane HospitalCostley Home, 1918GeographyLocationGreenlane, Auckland, New ZealandCoordinates36°53′36″S 174°46′49″E / 36.893429°S 174.780280°E / -36.893429; 174.780280OrganisationFundingPublic hospitalTypeGeneralHistoryFormer name(s)Costley Home for the Aged Poor, Auckland InfirmaryOpened1890LinksListsHospitals in New ...

Franck Solforosi

Questa voce sull'argomento canottieri francesi è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Franck SolforosiNazionalità Francia Altezza184 cm Peso70 kg Canottaggio CategoriaPesi leggeri Palmarès Competizione Ori Argenti Bronzi Giochi olimpici 0 0 1 Mondiali 2 2 1 Europei 0 1 2 Giochi del Mediterraneo 1 0 0 Vedi maggiori dettagliStatistiche aggiornate al 14 agosto 2016 Modifica dati su Wikidata · Manuale Franck Solforosi (Lione, 10 ...

Cathal Brugha

Irish revolutionary and republican politician (1874–1922) Cathal BrughaMinister for DefenceIn office1 April 1919 – 9 January 1922PresidentÉamon de ValeraPreceded byRichard MulcahySucceeded byRichard MulcahyCeann Comhairle of Dáil ÉireannIn office21 January 1919 – 22 January 1919DeputyJohn J. O'KellyPreceded byOffice establishedSucceeded byCount PlunkettPresident of Dáil ÉireannIn office21 January 1919 – 1 April 1919Preceded byNew officeSucceeded byÉamo...

Club Universitario de Deportes

Universitario de Deportes Généralités Nom complet Club Universitario de Deportes Surnoms La U Los MerenguesLos Cremas[1] Noms précédents Federación Universitaria de Deportes (1924-1933) Fondation 7 août 1924 (100 ans) Couleurs crème Stade Stade Monumental (80 093 places) Siège Av. Javier Prado Este,Cdra. 77 Ate, Lima Championnat actuel Liga 1 Président Jean Ferrari (administrateur provisoire) Entraîneur Fabian Bustos Joueur le plus capé José Luis Carranza (524) Me...

Philobryidae

Family of bivalves Philobryidae Philobrya modiolina Scientific classification Domain: Eukaryota Kingdom: Animalia Phylum: Mollusca Class: Bivalvia Order: Arcida Superfamily: Limopsoidea Family: PhilobryidaeBernardi, 1897 Genera 7, see text Philobryidae is a taxonomic family of very small equivalved triangular saltwater clams, marine bivalve molluscs, related to the ark clams and the bittersweets. Their shells have projecting umbones and are non-nacreous. The ligament of the hinge is either co...

/profillengkap.com/index.php/article/Actinic cheilitis

/profillengkap.com/index.php/en/Elizabeth (given name)

Isaac Kiese Thelin

Papirus 77

Schwäbisch Gmünd

Pembubaran Cekoslowakia

JBJ

Atanas Uzunov

العلاقات البنينية القبرصية

2018 USA Indoor Track and Field Championships

Élections législatives de 1986 en Martinique

دوري هونغ كونغ لكرة القدم 1986–87