. . . "Polinomios de Laguerre"@es . "\uB77C\uAC8C\uB974 \uB2E4\uD56D\uC2DD"@ko . . "LaguerrePolynomial"@en . . "Wong"@en . . . . . "28376"^^ . . . . . . . "\u5728\u6570\u5B66\u4E2D\uFF0C\u4EE5\u6CD5\u56FD\u6570\u5B66\u5BB6\u547D\u540D\u7684\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u5B9A\u4E49\u4E3A\u62C9\u76D6\u5C14\u65B9\u7A0B\u7684\u6807\u51C6\u89E3\u3002 \u8FD9\u662F\u4E00\u4E2A\u4E8C\u9636\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\u3002 \u8FD9\u4E2A\u65B9\u7A0B\u53EA\u6709\u5F53n\u975E\u8D1F\u65F6\uFF0C\u624D\u6709\u975E\u5947\u5F02\u89E3\u3002\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u53EF\u7528\u5728\u9AD8\u65AF\u79EF\u5206\u6CD5\u4E2D\uFF0C\u8BA1\u7B97\u5F62\u5982\u7684\u79EF\u5206\u3002 \u8FD9\u4E9B\u591A\u9879\u5F0F\uFF08\u901A\u5E38\u7528L0, L1\u7B49\u8868\u793A\uFF09\u6784\u6210\u4E00\u4E2A\u591A\u9879\u5F0F\u5E8F\u5217\u3002\u8FD9\u4E2A\u591A\u9879\u5F0F\u5E8F\u5217\u53EF\u4EE5\u7528\u7F57\u5FB7\u91CC\u683C\u516C\u5F0F\u9012\u63A8\u5F97\u5230\u3002 \u5728\u6309\u7167\u4E0B\u5F0F\u5B9A\u4E49\u7684\u5185\u79EF\u6784\u6210\u7684\u5185\u79EF\u7A7A\u95F4\u4E2D\uFF0C\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u662F\u6B63\u4EA4\u591A\u9879\u5F0F\u3002 \u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u6784\u6210\u4E00\u4E2A\u3002 \u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u5728\u91CF\u5B50\u529B\u5B66\u4E2D\u6709\u91CD\u8981\u5E94\u7528\u3002\u6C22\u539F\u5B50\u859B\u5B9A\u8C14\u65B9\u7A0B\u7684\u89E3\u7684\u5F84\u5411\u90E8\u5206\uFF0C\u5C31\u662F\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u3002 \u7269\u7406\u5B66\u5BB6\u901A\u5E38\u91C7\u7528\u53E6\u5916\u4E00\u79CD\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u7684\u5B9A\u4E49\u5F62\u5F0F\uFF0C\u5373\u5728\u4E0A\u9762\u7684\u5F62\u5F0F\u7684\u57FA\u7840\u4E0A\u4E58\u4E0A\u4E00\u4E2An!\u3002"@zh . "Koornwinder"@en . . "Laguerrepolynom \u00E4r ett matematiskt begrepp, d\u00E4r n te Laguerrepolynomet som svarar mot parametern , definierat enligt d\u00E4r \u00E4r ett reellt tal s\u00E5 att . F\u00F6r att f\u00F6lja den vanliga konventionen f\u00F6r definitionen av ortogonala polynom s\u00E5 kan man s\u00E4ga att Laguerrepolynomen svarar mot intervallet samt viktfunktionen. I viss litteratur f\u00F6rekommer ben\u00E4mningarna Laguerrepolynom samt generaliserade Laguerrepolynom f\u00F6r fallen respektive . Olikheten f\u00F6r parametern som f\u00F6rekommer i definitionen ovan, m\u00E5ste i allra h\u00F6gsta grad uppfyllas. F\u00F6r att f\u00F6rst\u00E5 n\u00F6dv\u00E4ndigheten i detta, f\u00F6ruts\u00E4tt f\u00F6r en stund att olikheten inte uppfylls. D\u00E5 kommer viktfunktionen inte vara integrerbar i origo, s\u00E5 att integralerna som definierar b\u00E5de ortogonalitet och norm f\u00F6r Laguerrepolynomen kommer att divergera. Laguerrepolynomen satisfierar Laguerreekvationen: Ett anv\u00E4ndningsomr\u00E5de f\u00F6r Laguerrepolynomen finns inom kvantmekaniken, d\u00E4r de f\u00F6rekommer d\u00E5 man behandlar v\u00E4teatomens tillst\u00E5nd. Laguerrepolynomen \u00E4r uppkallade efter Edmond Laguerre (1834-1886)."@sv . "Wielomiany Laguerre\u2019a \u2013 wielomiany o wsp\u00F3\u0142czynnikach rzeczywistych zdefiniowane jako:"@pl . . . . . . "Polin\u00F4mios de Laguerre"@pt . . . . "1121615716"^^ . "Polyn\u00F4me de Laguerre"@fr . . . . "In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834\u20131886), are solutions of Laguerre's equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of where n is still a non-negative integer.Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form These polynomials, usually denoted L0, L1, \u2026, are a polynomial sequence which may be defined by the Rodrigues formula, reducing to the closed form of a following section. They are orthogonal polynomials with respect to an inner product The sequence of Laguerre polynomials n! Ln is a Sheffer sequence, The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi\u2013Carlitz polynomials. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schr\u00F6dinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator. Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)"@en . . . . . . . . "\uC218\uD559\uC5D0\uC11C \uB77C\uAC8C\uB974 \uB2E4\uD56D\uC2DD(Laguerre\u591A\u9805\u5F0F, \uC601\uC5B4: Laguerre polynomial)\uC740 \uC9C1\uAD50 \uAD00\uACC4\uB97C \uB9CC\uC871\uC2DC\uD0A4\uB294 \uC77C\uB828\uC758 \uB2E4\uD56D\uC2DD\uB4E4\uC774\uB2E4. \uC591\uC790\uC5ED\uD559 \uB4F1\uC5D0\uC11C \uB4F1\uC7A5\uD55C\uB2E4."@ko . "Os polin\u00F4mios de Laguerre s\u00E3o uma fam\u00EDlia de em homenagem a Edmond Laguerre, e aparecem na an\u00E1lise de solu\u00E7\u00F5es para a equa\u00E7\u00E3o diferencial Desenvolvendo em s\u00E9rie de pot\u00EAncias, obtemos uma rela\u00E7\u00E3o de recorr\u00EAncia entre coeficientes consecutivos, Pode-se ver que quando n \u00E9 natural o coeficiente da pot\u00EAncia de grau maior e diferente de n se anula. Ou seja, uma solu\u00E7\u00E3o linearmente independente \u00E9 um polin\u00F4mio de grau n (polin\u00F4mio de Laguerre de ordem n, denotados por Ln(x)). Para encontrar a segunda solu\u00E7\u00E3o linearmente independente, deve-se estudar as solu\u00E7\u00F5es da equa\u00E7\u00E3o mais geral, que \u00E9 ."@pt . . . "In matematica, i polinomi di Laguerre, sono polinomi speciali costituenti una successione di polinomi, che hanno numerose applicazioni; il loro nome ricorda il matematico francese Edmond Nicolas Laguerre (1834-1886). Essi si possono definire con un'espressione alla Essi sono polinomi mutuamente ortogonali rispetto al prodotto interno espresso da La successione dei polinomi di Laguerre \u00E8 una sequenza di Sheffer."@it . . "In de wiskunde zijn de laguerre-polynomen, genoemd naar Edmond Laguerre (1834 - 1886), oplossingen van de -de differentiaalvergelijking van Laguerre: Laguerre-polynomen vinden toepassing in de kwantummechanica, in het radi\u00EBle deel van de oplossing van de schr\u00F6dingervergelijking voor een 1-elektron atoom."@nl . "Laguerre polynomials"@en . . . . "Os polin\u00F4mios de Laguerre s\u00E3o uma fam\u00EDlia de em homenagem a Edmond Laguerre, e aparecem na an\u00E1lise de solu\u00E7\u00F5es para a equa\u00E7\u00E3o diferencial Desenvolvendo em s\u00E9rie de pot\u00EAncias, obtemos uma rela\u00E7\u00E3o de recorr\u00EAncia entre coeficientes consecutivos, Pode-se ver que quando n \u00E9 natural o coeficiente da pot\u00EAncia de grau maior e diferente de n se anula. Ou seja, uma solu\u00E7\u00E3o linearmente independente \u00E9 um polin\u00F4mio de grau n (polin\u00F4mio de Laguerre de ordem n, denotados por Ln(x)). Para encontrar a segunda solu\u00E7\u00E3o linearmente independente, deve-se estudar as solu\u00E7\u00F5es da equa\u00E7\u00E3o mais geral, que \u00E9 ."@pt . . . "Orthogonal Polynomials"@en . . . . . "\u62C9\u76D6\u5C14\u591A\u9879\u5F0F"@zh . . "\u041C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B \u041B\u0430\u0433\u0435\u0440\u0440\u0430"@ru . . . . . . . . . . . "Laguerre polynomials"@en . . . . . "Los polinomios de Laguerre son una familia de polinomios ortogonales, llamados as\u00ED en honor de Edmond Laguerre, surgen al examinar las soluciones a la ecuaci\u00F3n diferencial: Desarrollando en serie de potencias se obtiene una relaci\u00F3n de recurrencia entre coeficientes consecutivos como la que sigue: Puede verse que siempre que n sea natural se anula el coeficiente de toda potencia mayor (y distinta) que n. Esto es, una de las soluciones linealmente independientes es un polinomio de grado n (polinomio de laguerre de orden n, que notaremos por Ln(x)). Para encontrar la otra soluci\u00F3n linealmente independiente han de estudiarse las soluciones de la ecuaci\u00F3n m\u00E1s general ."@es . . . . . . "Roelof"@en . . . "18"^^ . . . . "\u30E9\u30B2\u30FC\u30EB\u306E\u966A\u591A\u9805\u5F0F\uFF08\u30E9\u30B2\u30FC\u30EB\u306E\u3070\u3044\u305F\u3053\u3046\u3057\u304D\u3001associated Laguerre polynomials\uFF09\u3068\u306F\u3001\u5E38\u5FAE\u5206\u65B9\u7A0B\u5F0F \u3092\u6E80\u305F\u3059\u591A\u9805\u5F0F \u306E\u3053\u3068\u3092\u8A00\u3046\u3002\u305F\u3060\u3057 \u306F \u3092\u6E80\u305F\u3059\u6574\u6570\u3067\u3042\u308B\u3002 \u306E\u3068\u304D\u306E\u5FAE\u5206\u65B9\u7A0B\u5F0F\u306F\u30E9\u30B2\u30FC\u30EB\u306E\u5FAE\u5206\u65B9\u7A0B\u5F0F\u3068\u547C\u3070\u308C\u3001\u305D\u306E\u89E3 \u3092\u30E9\u30B2\u30FC\u30EB\u306E\u591A\u9805\u5F0F\u3068\u3044\u3046\u3002\u30E9\u30B2\u30FC\u30EB\u306E\u966A\u591A\u9805\u5F0F\u3068\u30E9\u30B2\u30FC\u30EB\u306E\u591A\u9805\u5F0F\u306F\u6B21\u306E\u95A2\u4FC2\u3067\u7D50\u3070\u308C\u3066\u3044\u308B\u3002 \u307E\u305F\u30ED\u30C9\u30EA\u30B2\u30B9\u306E\u516C\u5F0F (Rodrigues's Formula) \u3068\u3057\u3066\u4EE5\u4E0B\u306E\u5F62\u306B\u3082\u8868\u305B\u308B\u3002 \u6BCD\u95A2\u6570\u306F \u3067\u3042\u308B\u3002 \u306E\u3068\u304D\u306B\u3064\u3044\u3066 \u3068\u3044\u3046\u6F38\u5316\u5F0F\u304C\u6210\u308A\u7ACB\u3061\u3001\u5F8C\u8005\u304B\u3089 \u3067\u3042\u308B\u3002 \u91CF\u5B50\u529B\u5B66\u306B\u304A\u3044\u3066\u3001\u7403\u5BFE\u79F0\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\u306E\u30B7\u30E5\u30EC\u30C7\u30A3\u30F3\u30AC\u30FC\u65B9\u7A0B\u5F0F\uFF08\u4EE3\u8868\u7684\u306A\u3082\u306E\u306F\u6C34\u7D20\u539F\u5B50\u306B\u304A\u3051\u308B\u30B7\u30E5\u30EC\u30FC\u30C7\u30A3\u30F3\u30AC\u30FC\u65B9\u7A0B\u5F0F\uFF09\u306E\u52D5\u5F84\u65B9\u5411\u306E\u89E3\u306F\u3001\u30E9\u30B2\u30FC\u30EB\u306E\u966A\u591A\u9805\u5F0F\u3092\u7528\u3044\u3066\u8868\u3055\u308C\u308B\u3002"@ja . . . . . . . "Swarttouw"@en . "\u30E9\u30B2\u30FC\u30EB\u306E\u966A\u591A\u9805\u5F0F"@ja . . . "Laguerre-polynoom"@nl . "January 2016"@en . . . "Laguerre-Polynome (benannt nach Edmond Laguerre) sind spezielle Polynome, die auf dem Intervall ein orthogonales Funktionensystem bilden. Sie sind die L\u00F6sungen der laguerreschen Differentialgleichung. Eine wichtige Rolle spielen die Laguerre-Polynome in der theoretischen Physik, insbesondere in der Quantenmechanik."@de . . . . "Polinomial laguerre"@in . . . . "Laguerre-Polynome"@de . . "\u041F\u043E\u043B\u0456\u043D\u043E\u043C\u0438 \u041B\u0430\u0491\u0435\u0440\u0440\u0430 \u2014 \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u0456 \u043F\u043E\u043B\u0456\u043D\u043E\u043C\u0438, \u043D\u0430\u0437\u0432\u0430\u043D\u0456 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0444\u0440\u0430\u043D\u0446\u0443\u0437\u044C\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u0415\u0434\u043C\u043E\u043D\u0430 \u041B\u0430\u0491\u0435\u0440\u0440\u0430."@uk . "Koekoek"@en . "En math\u00E9matiques, les polyn\u00F4mes de Laguerre, nomm\u00E9s d'apr\u00E8s Edmond Laguerre, sont les solutions normalis\u00E9es de l'\u00E9quation de Laguerre : qui est une \u00E9quation diff\u00E9rentielle lin\u00E9aire homog\u00E8ne d'ordre 2 et se r\u00E9\u00E9crit sous la forme de Sturm-Liouville : Cette \u00E9quation a des solutions non singuli\u00E8res seulement si n est un entier positif.Les solutions Ln forment une suite de polyn\u00F4mes orthogonaux dans L2 (\u211D+, e\u2013xdx), et la normalisation se fait en leur imposant d'\u00EAtre de norme 1, donc de former une famille orthonormale. Ils forment m\u00EAme une base hilbertienne de L2(\u211D+, e\u2013xdx)."@fr . . . "\u0412 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B \u041B\u0430\u0433\u0435\u0301\u0440\u0440\u0430, \u043D\u0430\u0437\u0432\u0430\u043D\u043D\u044B\u0435 \u0432 \u0447\u0435\u0441\u0442\u044C \u042D\u0434\u043C\u043E\u043D\u0430 \u041B\u0430\u0433\u0435\u0440\u0440\u0430 (1834\u20141886),\u044F\u0432\u043B\u044F\u044E\u0442\u0441\u044F \u043A\u0430\u043D\u043E\u043D\u0438\u0447\u0435\u0441\u043A\u0438\u043C\u0438 \u0440\u0435\u0448\u0435\u043D\u0438\u044F\u043C\u0438 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u044F \u041B\u0430\u0433\u0435\u0440\u0440\u0430: \u044F\u0432\u043B\u044F\u044E\u0449\u0435\u0433\u043E\u0441\u044F \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u043C \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u043C \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435\u043C \u0432\u0442\u043E\u0440\u043E\u0433\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0430. \u0412 \u0444\u0438\u0437\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043A\u0438\u043D\u0435\u0442\u0438\u043A\u0435 \u044D\u0442\u0438 \u0436\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B (\u0438\u043D\u043E\u0433\u0434\u0430 \u0441 \u0442\u043E\u0447\u043D\u043E\u0441\u0442\u044C\u044E \u0434\u043E \u043D\u043E\u0440\u043C\u0438\u0440\u043E\u0432\u043A\u0438) \u043F\u0440\u0438\u043D\u044F\u0442\u043E \u043D\u0430\u0437\u044B\u0432\u0430\u0442\u044C \u043F\u043E\u043B\u0438\u043D\u043E\u043C\u0430\u043C\u0438 \u0421\u043E\u043D\u0438\u043D\u0430 \u0438\u043B\u0438 \u0421\u043E\u043D\u0438\u043D\u0430 \u2014 \u041B\u0430\u0433\u0435\u0440\u0440\u0430. \u041C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B \u041B\u0430\u0433\u0435\u0440\u0440\u0430 \u0442\u0430\u043A\u0436\u0435 \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u0443\u044E\u0442\u0441\u044F \u0432 \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u0443\u0440\u043D\u043E\u0439 \u0444\u043E\u0440\u043C\u0443\u043B\u0435 \u0413\u0430\u0443\u0441\u0441\u0430 \u2014 \u041B\u0430\u0433\u0435\u0440\u0440\u0430 \u0447\u0438\u0441\u043B\u0435\u043D\u043D\u043E\u0433\u043E \u0432\u044B\u0447\u0438\u0441\u043B\u0435\u043D\u0438\u044F \u0438\u043D\u0442\u0435\u0433\u0440\u0430\u043B\u043E\u0432 \u0432\u0438\u0434\u0430: \u041C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B \u041B\u0430\u0433\u0435\u0440\u0440\u0430, \u043E\u0431\u044B\u0447\u043D\u043E \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u044E\u0449\u0438\u0435\u0441\u044F \u043A\u0430\u043A , \u044F\u0432\u043B\u044F\u044E\u0442\u0441\u044F \u043F\u043E\u0441\u043B\u0435\u0434\u043E\u0432\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u0441\u0442\u044C\u044E \u043F\u043E\u043B\u0438\u043D\u043E\u043C\u043E\u0432, \u043A\u043E\u0442\u043E\u0440\u0430\u044F \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u043D\u0430\u0439\u0434\u0435\u043D\u0430 \u043F\u043E \u0444\u043E\u0440\u043C\u0443\u043B\u0435 \u0420\u043E\u0434\u0440\u0438\u0433\u0430 \u042D\u0442\u0438 \u043F\u043E\u043B\u0438\u043D\u043E\u043C\u044B \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u044B \u0434\u0440\u0443\u0433 \u0434\u0440\u0443\u0433\u0443 \u0441\u043E \u0441\u043A\u0430\u043B\u044F\u0440\u043D\u044B\u043C \u043F\u0440\u043E\u0438\u0437\u0432\u0435\u0434\u0435\u043D\u0438\u0435\u043C: \u041F\u043E\u0441\u043B\u0435\u0434\u043E\u0432\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u0441\u0442\u044C \u043F\u043E\u043B\u0438\u043D\u043E\u043C\u043E\u0432 \u041B\u0430\u0433\u0435\u0440\u0440\u0430 \u2014 \u044D\u0442\u043E . \u041C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B \u041B\u0430\u0433\u0435\u0440\u0440\u0430 \u043F\u0440\u0438\u043C\u0435\u043D\u044F\u044E\u0442\u0441\u044F \u0432 \u043A\u0432\u0430\u043D\u0442\u043E\u0432\u043E\u0439 \u043C\u0435\u0445\u0430\u043D\u0438\u043A\u0435, \u0432 \u0440\u0430\u0434\u0438\u0430\u043B\u044C\u043D\u043E\u0439 \u0447\u0430\u0441\u0442\u0438 \u0440\u0435\u0448\u0435\u043D\u0438\u044F \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u044F \u0428\u0440\u0451\u0434\u0438\u043D\u0433\u0435\u0440\u0430 \u0434\u043B\u044F \u0430\u0442\u043E\u043C\u0430 \u0441 \u043E\u0434\u043D\u0438\u043C \u044D\u043B\u0435\u043A\u0442\u0440\u043E\u043D\u043E\u043C. \u0418\u043C\u0435\u044E\u0442\u0441\u044F \u0438 \u0434\u0440\u0443\u0433\u0438\u0435 \u043F\u0440\u0438\u043C\u0435\u043D\u0435\u043D\u0438\u044F \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u043E\u0432 \u041B\u0430\u0433\u0435\u0440\u0440\u0430."@ru . "943917"^^ . . . "p/l057310"@en . "Laguerrovy polynomy, pojmenovan\u00E9 po (1834 \u2013 1886), je jeden z ortogon\u00E1ln\u00EDch syst\u00E9m\u016F polynom\u016F. Vyu\u017E\u00EDvaj\u00ED se nap\u0159\u00EDklad v kvantov\u00E9 mechanice pro popis vlnov\u00E9 funkce odpov\u00EDdaj\u00EDc\u00ED stav\u016Fm atomu vod\u00EDku."@cs . . "Laguerre-Polynome (benannt nach Edmond Laguerre) sind spezielle Polynome, die auf dem Intervall ein orthogonales Funktionensystem bilden. Sie sind die L\u00F6sungen der laguerreschen Differentialgleichung. Eine wichtige Rolle spielen die Laguerre-Polynome in der theoretischen Physik, insbesondere in der Quantenmechanik."@de . . . "Laguerrovy polynomy"@cs . "In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834\u20131886), are solutions of Laguerre's equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of where n is still a non-negative integer.Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin)."@en . . "\u041F\u043E\u043B\u0456\u043D\u043E\u043C\u0438 \u041B\u0430\u0491\u0435\u0440\u0440\u0430"@uk . . "\u041F\u043E\u043B\u0456\u043D\u043E\u043C\u0438 \u041B\u0430\u0491\u0435\u0440\u0440\u0430 \u2014 \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u0456 \u043F\u043E\u043B\u0456\u043D\u043E\u043C\u0438, \u043D\u0430\u0437\u0432\u0430\u043D\u0456 \u043D\u0430 \u0447\u0435\u0441\u0442\u044C \u0444\u0440\u0430\u043D\u0446\u0443\u0437\u044C\u043A\u043E\u0433\u043E \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0430 \u0415\u0434\u043C\u043E\u043D\u0430 \u041B\u0430\u0491\u0435\u0440\u0440\u0430."@uk . "\u5728\u6570\u5B66\u4E2D\uFF0C\u4EE5\u6CD5\u56FD\u6570\u5B66\u5BB6\u547D\u540D\u7684\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u5B9A\u4E49\u4E3A\u62C9\u76D6\u5C14\u65B9\u7A0B\u7684\u6807\u51C6\u89E3\u3002 \u8FD9\u662F\u4E00\u4E2A\u4E8C\u9636\u7EBF\u6027\u5FAE\u5206\u65B9\u7A0B\u3002 \u8FD9\u4E2A\u65B9\u7A0B\u53EA\u6709\u5F53n\u975E\u8D1F\u65F6\uFF0C\u624D\u6709\u975E\u5947\u5F02\u89E3\u3002\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u53EF\u7528\u5728\u9AD8\u65AF\u79EF\u5206\u6CD5\u4E2D\uFF0C\u8BA1\u7B97\u5F62\u5982\u7684\u79EF\u5206\u3002 \u8FD9\u4E9B\u591A\u9879\u5F0F\uFF08\u901A\u5E38\u7528L0, L1\u7B49\u8868\u793A\uFF09\u6784\u6210\u4E00\u4E2A\u591A\u9879\u5F0F\u5E8F\u5217\u3002\u8FD9\u4E2A\u591A\u9879\u5F0F\u5E8F\u5217\u53EF\u4EE5\u7528\u7F57\u5FB7\u91CC\u683C\u516C\u5F0F\u9012\u63A8\u5F97\u5230\u3002 \u5728\u6309\u7167\u4E0B\u5F0F\u5B9A\u4E49\u7684\u5185\u79EF\u6784\u6210\u7684\u5185\u79EF\u7A7A\u95F4\u4E2D\uFF0C\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u662F\u6B63\u4EA4\u591A\u9879\u5F0F\u3002 \u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u6784\u6210\u4E00\u4E2A\u3002 \u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u5728\u91CF\u5B50\u529B\u5B66\u4E2D\u6709\u91CD\u8981\u5E94\u7528\u3002\u6C22\u539F\u5B50\u859B\u5B9A\u8C14\u65B9\u7A0B\u7684\u89E3\u7684\u5F84\u5411\u90E8\u5206\uFF0C\u5C31\u662F\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u3002 \u7269\u7406\u5B66\u5BB6\u901A\u5E38\u91C7\u7528\u53E6\u5916\u4E00\u79CD\u62C9\u76D6\u5C14\u591A\u9879\u5F0F\u7684\u5B9A\u4E49\u5F62\u5F0F\uFF0C\u5373\u5728\u4E0A\u9762\u7684\u5F62\u5F0F\u7684\u57FA\u7840\u4E0A\u4E58\u4E0A\u4E00\u4E2An!\u3002"@zh . . . "En math\u00E9matiques, les polyn\u00F4mes de Laguerre, nomm\u00E9s d'apr\u00E8s Edmond Laguerre, sont les solutions normalis\u00E9es de l'\u00E9quation de Laguerre : qui est une \u00E9quation diff\u00E9rentielle lin\u00E9aire homog\u00E8ne d'ordre 2 et se r\u00E9\u00E9crit sous la forme de Sturm-Liouville : Cette \u00E9quation a des solutions non singuli\u00E8res seulement si n est un entier positif.Les solutions Ln forment une suite de polyn\u00F4mes orthogonaux dans L2 (\u211D+, e\u2013xdx), et la normalisation se fait en leur imposant d'\u00EAtre de norme 1, donc de former une famille orthonormale. Ils forment m\u00EAme une base hilbertienne de L2(\u211D+, e\u2013xdx). Cette suite de polyn\u00F4mes peut \u00EAtre d\u00E9finie par la formule de Rodrigues La suite des polyn\u00F4mes de Laguerre est une suite de Sheffer. Les polyn\u00F4mes de Laguerre apparaissent en m\u00E9canique quantique dans la partie radiale de la solution de l'\u00E9quation de Schr\u00F6dinger pour un atome \u00E0 un \u00E9lectron. Le coefficient dominant de Ln est (\u20131)n/n!. Les physiciens utilisent souvent une d\u00E9finition des polyn\u00F4mes de Laguerre o\u00F9 ceux-ci sont multipli\u00E9s par (\u20131)nn!, obtenant ainsi des polyn\u00F4mes unitaires."@fr . "In matematica, i polinomi di Laguerre, sono polinomi speciali costituenti una successione di polinomi, che hanno numerose applicazioni; il loro nome ricorda il matematico francese Edmond Nicolas Laguerre (1834-1886). Essi si possono definire con un'espressione alla Essi sono polinomi mutuamente ortogonali rispetto al prodotto interno espresso da La successione dei polinomi di Laguerre \u00E8 una sequenza di Sheffer."@it . . . . . "\uC218\uD559\uC5D0\uC11C \uB77C\uAC8C\uB974 \uB2E4\uD56D\uC2DD(Laguerre\u591A\u9805\u5F0F, \uC601\uC5B4: Laguerre polynomial)\uC740 \uC9C1\uAD50 \uAD00\uACC4\uB97C \uB9CC\uC871\uC2DC\uD0A4\uB294 \uC77C\uB828\uC758 \uB2E4\uD56D\uC2DD\uB4E4\uC774\uB2E4. \uC591\uC790\uC5ED\uD559 \uB4F1\uC5D0\uC11C \uB4F1\uC7A5\uD55C\uB2E4."@ko . . . . . . . . "Tom H."@en . . . "Wielomiany Laguerre\u2019a"@pl . "Limit as n goes to infinity?"@en . . "Laguerre polynomial"@en . "Laguerrovy polynomy, pojmenovan\u00E9 po (1834 \u2013 1886), je jeden z ortogon\u00E1ln\u00EDch syst\u00E9m\u016F polynom\u016F. Vyu\u017E\u00EDvaj\u00ED se nap\u0159\u00EDklad v kvantov\u00E9 mechanice pro popis vlnov\u00E9 funkce odpov\u00EDdaj\u00EDc\u00ED stav\u016Fm atomu vod\u00EDku."@cs . . "Wielomiany Laguerre\u2019a \u2013 wielomiany o wsp\u00F3\u0142czynnikach rzeczywistych zdefiniowane jako:"@pl . "Laguerrepolynom \u00E4r ett matematiskt begrepp, d\u00E4r n te Laguerrepolynomet som svarar mot parametern , definierat enligt d\u00E4r \u00E4r ett reellt tal s\u00E5 att . F\u00F6r att f\u00F6lja den vanliga konventionen f\u00F6r definitionen av ortogonala polynom s\u00E5 kan man s\u00E4ga att Laguerrepolynomen svarar mot intervallet samt viktfunktionen. I viss litteratur f\u00F6rekommer ben\u00E4mningarna Laguerrepolynom samt generaliserade Laguerrepolynom f\u00F6r fallen respektive . Laguerrepolynomen satisfierar Laguerreekvationen: Laguerrepolynomen \u00E4r uppkallade efter Edmond Laguerre (1834-1886)."@sv . . "Los polinomios de Laguerre son una familia de polinomios ortogonales, llamados as\u00ED en honor de Edmond Laguerre, surgen al examinar las soluciones a la ecuaci\u00F3n diferencial: Desarrollando en serie de potencias se obtiene una relaci\u00F3n de recurrencia entre coeficientes consecutivos como la que sigue:"@es . . "Roderick S. C."@en . . . . "In de wiskunde zijn de laguerre-polynomen, genoemd naar Edmond Laguerre (1834 - 1886), oplossingen van de -de differentiaalvergelijking van Laguerre: Laguerre-polynomen vinden toepassing in de kwantummechanica, in het radi\u00EBle deel van de oplossing van de schr\u00F6dingervergelijking voor een 1-elektron atoom."@nl . . "\u0412 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B \u041B\u0430\u0433\u0435\u0301\u0440\u0440\u0430, \u043D\u0430\u0437\u0432\u0430\u043D\u043D\u044B\u0435 \u0432 \u0447\u0435\u0441\u0442\u044C \u042D\u0434\u043C\u043E\u043D\u0430 \u041B\u0430\u0433\u0435\u0440\u0440\u0430 (1834\u20141886),\u044F\u0432\u043B\u044F\u044E\u0442\u0441\u044F \u043A\u0430\u043D\u043E\u043D\u0438\u0447\u0435\u0441\u043A\u0438\u043C\u0438 \u0440\u0435\u0448\u0435\u043D\u0438\u044F\u043C\u0438 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u044F \u041B\u0430\u0433\u0435\u0440\u0440\u0430: \u044F\u0432\u043B\u044F\u044E\u0449\u0435\u0433\u043E\u0441\u044F \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u043C \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u043C \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0435\u043C \u0432\u0442\u043E\u0440\u043E\u0433\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0430. \u0412 \u0444\u0438\u0437\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043A\u0438\u043D\u0435\u0442\u0438\u043A\u0435 \u044D\u0442\u0438 \u0436\u0435 \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B (\u0438\u043D\u043E\u0433\u0434\u0430 \u0441 \u0442\u043E\u0447\u043D\u043E\u0441\u0442\u044C\u044E \u0434\u043E \u043D\u043E\u0440\u043C\u0438\u0440\u043E\u0432\u043A\u0438) \u043F\u0440\u0438\u043D\u044F\u0442\u043E \u043D\u0430\u0437\u044B\u0432\u0430\u0442\u044C \u043F\u043E\u043B\u0438\u043D\u043E\u043C\u0430\u043C\u0438 \u0421\u043E\u043D\u0438\u043D\u0430 \u0438\u043B\u0438 \u0421\u043E\u043D\u0438\u043D\u0430 \u2014 \u041B\u0430\u0433\u0435\u0440\u0440\u0430. \u041C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B \u041B\u0430\u0433\u0435\u0440\u0440\u0430 \u0442\u0430\u043A\u0436\u0435 \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u0443\u044E\u0442\u0441\u044F \u0432 \u043A\u0432\u0430\u0434\u0440\u0430\u0442\u0443\u0440\u043D\u043E\u0439 \u0444\u043E\u0440\u043C\u0443\u043B\u0435 \u0413\u0430\u0443\u0441\u0441\u0430 \u2014 \u041B\u0430\u0433\u0435\u0440\u0440\u0430 \u0447\u0438\u0441\u043B\u0435\u043D\u043D\u043E\u0433\u043E \u0432\u044B\u0447\u0438\u0441\u043B\u0435\u043D\u0438\u044F \u0438\u043D\u0442\u0435\u0433\u0440\u0430\u043B\u043E\u0432 \u0432\u0438\u0434\u0430: \u041C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u044B \u041B\u0430\u0433\u0435\u0440\u0440\u0430, \u043E\u0431\u044B\u0447\u043D\u043E \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u044E\u0449\u0438\u0435\u0441\u044F \u043A\u0430\u043A , \u044F\u0432\u043B\u044F\u044E\u0442\u0441\u044F \u043F\u043E\u0441\u043B\u0435\u0434\u043E\u0432\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u0441\u0442\u044C\u044E \u043F\u043E\u043B\u0438\u043D\u043E\u043C\u043E\u0432, \u043A\u043E\u0442\u043E\u0440\u0430\u044F \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u043D\u0430\u0439\u0434\u0435\u043D\u0430 \u043F\u043E \u0444\u043E\u0440\u043C\u0443\u043B\u0435 \u0420\u043E\u0434\u0440\u0438\u0433\u0430"@ru . . . . . "\u30E9\u30B2\u30FC\u30EB\u306E\u966A\u591A\u9805\u5F0F\uFF08\u30E9\u30B2\u30FC\u30EB\u306E\u3070\u3044\u305F\u3053\u3046\u3057\u304D\u3001associated Laguerre polynomials\uFF09\u3068\u306F\u3001\u5E38\u5FAE\u5206\u65B9\u7A0B\u5F0F \u3092\u6E80\u305F\u3059\u591A\u9805\u5F0F \u306E\u3053\u3068\u3092\u8A00\u3046\u3002\u305F\u3060\u3057 \u306F \u3092\u6E80\u305F\u3059\u6574\u6570\u3067\u3042\u308B\u3002 \u306E\u3068\u304D\u306E\u5FAE\u5206\u65B9\u7A0B\u5F0F\u306F\u30E9\u30B2\u30FC\u30EB\u306E\u5FAE\u5206\u65B9\u7A0B\u5F0F\u3068\u547C\u3070\u308C\u3001\u305D\u306E\u89E3 \u3092\u30E9\u30B2\u30FC\u30EB\u306E\u591A\u9805\u5F0F\u3068\u3044\u3046\u3002\u30E9\u30B2\u30FC\u30EB\u306E\u966A\u591A\u9805\u5F0F\u3068\u30E9\u30B2\u30FC\u30EB\u306E\u591A\u9805\u5F0F\u306F\u6B21\u306E\u95A2\u4FC2\u3067\u7D50\u3070\u308C\u3066\u3044\u308B\u3002 \u307E\u305F\u30ED\u30C9\u30EA\u30B2\u30B9\u306E\u516C\u5F0F (Rodrigues's Formula) \u3068\u3057\u3066\u4EE5\u4E0B\u306E\u5F62\u306B\u3082\u8868\u305B\u308B\u3002 \u6BCD\u95A2\u6570\u306F \u3067\u3042\u308B\u3002 \u306E\u3068\u304D\u306B\u3064\u3044\u3066 \u3068\u3044\u3046\u6F38\u5316\u5F0F\u304C\u6210\u308A\u7ACB\u3061\u3001\u5F8C\u8005\u304B\u3089 \u3067\u3042\u308B\u3002 \u91CF\u5B50\u529B\u5B66\u306B\u304A\u3044\u3066\u3001\u7403\u5BFE\u79F0\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\u306E\u30B7\u30E5\u30EC\u30C7\u30A3\u30F3\u30AC\u30FC\u65B9\u7A0B\u5F0F\uFF08\u4EE3\u8868\u7684\u306A\u3082\u306E\u306F\u6C34\u7D20\u539F\u5B50\u306B\u304A\u3051\u308B\u30B7\u30E5\u30EC\u30FC\u30C7\u30A3\u30F3\u30AC\u30FC\u65B9\u7A0B\u5F0F\uFF09\u306E\u52D5\u5F84\u65B9\u5411\u306E\u89E3\u306F\u3001\u30E9\u30B2\u30FC\u30EB\u306E\u966A\u591A\u9805\u5F0F\u3092\u7528\u3044\u3066\u8868\u3055\u308C\u308B\u3002"@ja . "Laguerrepolynom"@sv . . . . "Polinomi di Laguerre"@it . . "Ren\u00E9 F."@en . . . . .