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Topic: Laguerre polynomials

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	PlanetMath: Laguerre polynomial
		The ordinary Laguerre polynomials are the special case of the generalized Laguerre polynomials when the parameter goes to zero.
		When some result holds for generalized Laguerre polynomials which is not more complicated than that for ordinary Laguerre polynomials, we shall only provide the more general result and leave it to the reader to send the parameter to zero to recover the more specific result.
		This is version 17 of Laguerre polynomial, born on 2002-03-06, modified 2007-03-03.
	planetmath.org /encyclopedia/LaguerrePolynomial.html (323 words)

	Laguerre (print-only)
		Laguerre studied approximation methods and is best remembered for the special functions the Laguerre polynomials which are solutions of the Laguerre differential equations.
		Laguerre was pictured by his contemporaries as a quiet, gentle man who was passionately devoted to his research, his teaching, and the education of his two daughters.
		Laguerre wrote 140 memoirs which he published in the leading journals of his time so it is reasonable to ask why he is only known for the results mentioned specifically above.
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Laguerre.html (912 words)

	Orthogonal polynomials - Wikipedia, the free encyclopedia
		The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Stieltjes.
		Because all polynomial sequences arising from a differential equation in the manner described above are trivially equivalent to the classical polynomials, the actual classical polynomials are always used.
		The Jacobi-like polynomials, once they have had their domain shifted and scaled so that the interval of orthogonality is [-1, 1], still have two parameters to be determined.
	en.wikipedia.org /wiki/Orthogonal_polynomials (1945 words)

	Laguerre polynomials - Wikipedia, the free encyclopedia
		The polynomials may be expressed in terms of a contour integral
		The generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator, due to their relation to the Hermite polynomials, which can be expressed as
		The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as
	en.wikipedia.org /wiki/Laguerre_polynomials (306 words)

	Laguerre Polynomials
		The Laguerre polynomials are orthogonal on the interval from 0 to ∞ with respect to the weight function w(x) = e
		The differential equation is not affected by changes in a multiplicative constant in the definition of the polynomial.
		The Laguerre polynomials are really rather straightforward examples of orthogonal polynomials, and most of their properties can be derived from Rodrigues' formula.
	www.du.edu /%7ejcalvert/math/laguerre.htm (972 words)

	Laguerre Polynomials in a Relativistic Quantum-Statistical Model (ResearchIndex)
		Laguerre Polynomials in a Relativistic Quantum-Statistical Model (1996)
		It is shown that the Laguerre polynomials depending on the effective charge are solutions for self-consistent fields.
		0.2: The q-analogue of the dual Hahn Polynomials: Its..
	citeseer.ist.psu.edu /65738.html (357 words)

	CiteULike: Specializations of generalized Laguerre polynomials (Site not responding. Last check: 2007-10-18)
		A classical result relating weighted Motzkin paths, moment sequences of orthogonal polynomial families (o.p.f.), and continued fractions states that the moments $μ_n$ of the o.p.f.
		In this paper the particular choice of $b_k=a[k+1]_{r,s}+b[k]_{t,u}$, $\lambda_k=ab[k]_{p,q}[k]_{v,w}$ is considered, where $[k]_{r,s}=(r^k-s^k)/(r-s)$, etc. A general (bijective) result shows how the moments $\mu_n$ can be interpreted as generating polynomials for certain statistics on permutations associated with the 8 counting parameters $p,q,r,s,t,u,v,\break w$.
		These cases are related to the little $q$-Jacobi polynomials and to the $q$-Laguerre polynomials.}, author = {Simion, R. and Stanton, D. citeulike-article-id = {671653}, doi = {10.1137/S003614109322854X}, journal = {SIAM J. Math.
	www.citeulike.org /user/dsquared/article/671653 (337 words)

	Reading Class: Hecke Algebras and Orthogonal Polynomials
		Macdonald polynomials as simultaneous eigenfunctions of operators and existence proof.
		Macdonald polynomials as simultaneous eigenfunctions of Dunkl-like operators in the double affine Hecke algebra.
		Gus on Jack and Macdonald polynomials as orthogonal polys
	www.math.ucdavis.edu /~vazirani/W05/290.html (474 words)

	question on Laguerre polynomials of matrix argument (Site not responding. Last check: 2007-10-18)
		Subject: question on Laguerre polynomials of matrix argument
		Tom Koornwinder -------------------------------------------------------- What is the kernel function for the Laguerre polynomials when the variables are 2 by 2 hermitian matrices, rather than real?
		The question may come down to expressing a Bessel function of a matrix (the 2 by 2 hermitian case is what I have) in a nice closed form in terms of ordinary scalar Bessel functions, if that is possible.
	cio.nist.gov /esd/emaildir/lists/opsftalk/msg00030.html (235 words)

	Transactions of the American Mathematical Society
		the polynomials are eigenfunctions of a unique differential operator of the form
		I.H. Jung, K.H. Kwon, D.W. Lee and L.L. Littlejohn : Sobolev orthogonal polynomials and spectral differential equations.
		Srinivasa Rao, R. Jagannathan, G. Vanden Berghe, J. Van der Jeugt (ed.),Differential operators having Laguerre type and Sobolev type Laguerre polynomials as eigenfunctions: a survey,Special Functions and Differential Equations (Madras, India, January 13-24, 1997), Allied Publishers Private Lt., New Delhi, India, 1998, pp.
	www.ams.org /joursearch/servlet/DoSearch?f1=msc&v1=34A35&jrnl=one&onejrnl=tran (601 words)

	Citebase - Asymptotic zero behavior of Laguerre polynomials with negative parameter (Site not responding. Last check: 2007-10-18)
		We use a Riemann-Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.
		We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthog...
		We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [-1,1].
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0205175 (1481 words)

	Proceedings of the American Mathematical Society
		It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order
		J. Koekoek and R. Koekoek: On a differential equation for Koornwinder's generalized Laguerre polynomials.
		J. Koekoek, R. Koekoek and H. Bavinck: On differential equations for Sobolev-type Laguerre polynomials.
	www.ams.org /proc/1997-125-12/S0002-9939-97-04091-4/home.html (356 words)

	Laguerre Polynomial
		In the Sturm-Liouville Boundary Value Problem, there is a special case called Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics.
		Generating Function: The generating function of a Laguerre Polynomial is:
		Recurrence Relation: A Laguerre Polynomial at one point can be expressed in terms of neighboring Laguerre Polynomials at the same point.
	www.efunda.com /math/Laguerre/index.cfm (124 words)

	Orthogonal Polynomials
		Section 3.2.10 discusses the generalization of Legendre polynomials to Legendre functions, which can have non-integer degrees.
		Series of Chebyshev polynomials are often used in making numerical approximations to functions.
		Legendre, Gegenbauer and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials.
	documents.wolfram.com /v4/MainBook/3.2.9.html (240 words)

	Laguerre Polynomials
		Here are the first seven Laguerre Polynomials, along with a plot.
		The eigenvalues of Laguerre's equation are the nonnegative integers n.
		By one of the major Sturm-Liouville theorems, the Laguerre polynomials are orthogonal with respect to the weight Exp[-x].
	www.ma.iup.edu /projects/CalcDEMma/sturm/sl010.html (128 words)

	The Legendre and Laguerre Polynomials & the Elementary Quantum Mechanical Model of the Hydrogen Atom
		The Legendre and Laguerre Polynomials and the Elementary Quantum Mechanical Model of the Hydrogen Atom
		In this report, we explore the basic Quantum Mechanical analysis of hydrogen.
		In the process we come across the Legendre and Laguerre polynomials, and present an informal derivation of these functions and their normalization.
	www.physics.drexel.edu /~tim/open/hydrofin (64 words)

	Citebase - Specializations of generalized Laguerre polynomials (Site not responding. Last check: 2007-10-18)
		Authors: Simion, Rodica; Stanton, Dennis W. Three specializations of a set of orthogonal polynomials with ``8 different q's'' are given.
		Use the Correlation Generator to explore the correlation between download impact ("hits") and citation impact.
		The polynomials turn out to be a sum of two little q-Jacobi polynomials, q-n-2 0
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9307219 (468 words)

	Citebase - q-Laguerre polynomials and big q-Bessel functions and their orthogonality relations (Site not responding. Last check: 2007-10-18)
		D.S. Moak, The q-analogue of the Laguerre polynomials, J. Math.
		In this paper we present an addition to Askey's scheme of q-hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only.
		This paper studies properties of q-Jacobi polynomials and their duals by means of operators of the discrete series representations for the quantum algebra U
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/9805023 (1336 words)

	Specializations of Generalized Laguerre Polynomials
		Specializations of Generalized Laguerre Polynomials:SIAM Journal on Mathematical Analysis Vol.
		Three specializations of a set of orthogonal polynomials with "8 different q's" are given.
		The polynomials are identified as q-analogues of Laguerre polynomials, and the combinatorial interpretation of the moments gives infinitely many new Mahonian statistics on permutations.
	epubs.siam.org /sam-bin/dbq/article/22854 (66 words)

	Bibliography
		Cody, A survey of practical rational and polynomial approximation of functions, SIAM Rev. 12 (1970), 400-423.
		Gabutti and B. Minetti, A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function, J. Comput.
		Pabon-Ortiz and M. Artoni, Laguerre polynomials: Novel properties and numerical generation scheme for analysis of a discrete probability distribution, Comput.
	math.nist.gov /mcsd/Reports/2001/nesf/node38.html (8398 words)

	GiNaC is Not a CAS
		Its design is revolutionary in a sense that contrary to other CAS it does not try to provide extensive algebraic capabilities and a simple programming language but instead accepts a given language (C++) and extends it by a set of algebraic capabilities.
		(z) to compute the first few Laguerre polynomials in the symbolic variable x.
		Very fast manipulation of large integers and rationals owing to the CLN library (for instance, it uses Karatsuba multiplication and Schönhage-Strassen multiplication for very large integers).
	www.ginac.de /About.html (318 words)

	Citebase - Integral representations for multiple Hermite and multiple Laguerre polynomials (Site not responding. Last check: 2007-10-18)
		Authors: Bleher, Pavel M. Kuijlaars, Arno B. We give integral representations for multiple Hermite and multiple Hermite polynomials of both type I and II.
		We also show how these are connected with double integral representations of certain kernels from random matrix theory.
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0406616 (916 words)

	Fractional Derivative of Riemann-Liouville via Laguerre Polynomials
		Abstract.- We show that well known properties of Laguerre polynomials permit to motivate the definition of Riemann-Liouville for the fractional derivative.
		The associated Laguerre polynomials are given by [1]:
		In this work we accept that (1) is valid for
	www.wbabin.net /bonilla/fractional.htm (222 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations (Site not responding. Last check: 2007-10-18)
		Energy Citations Database (ECD) Document #7235773 - A new asymptotic form for the Laguerre polynomials
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		A new asymptotic form for the Laguerre polynomials
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=7235773 (107 words)

	Series of Legendre and Laguerre polynomials, V. F. Cowling
		Series of Legendre and Laguerre polynomials, V. Cowling
		If you would like to purchase this specific document for $25, please click the "buy article" button below.
		Please recommend that your library subscribe to DMJ and DMJ 100 by printing one or both of the following forms and submitting them directly to your library.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1077467786 (71 words)

	Math Forum Discussions - Laplace limit using Laguerre polynomials
		Math Forum Discussions - Laplace limit using Laguerre polynomials
		formulation using associated Laguerre polynomials which makes the
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?forumID=13&threadID=1428699&messageID=5005457 (158 words)

	POLPAK - Recursive Polynomials
		It includes routines to evaluate the recursively defined polynomial families of
		A variety of other polynomials and functions have been added.
		bpab.m, evaluates the Bernstein polynomials defined on the interval [A,B];
	www.csit.fsu.edu /~burkardt/m_src/polpak/polpak.html (868 words)

	RMC - Vol. 19, Num. 2 - Kruglyak et al.
		Kruglyak, L. Maligranda, and L.-E. Persson, Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation, Rev. Mat.
		Structure of the Hardy Operator Related to Laguerre Polynomials and the Euler Dierential Equation
		Key words: Hardy inequality, Hardy operator, Laguerre polynomials, isometry, Lebesgue spaces, basis in
	www.mat.ucm.es /serv/revista/vol19-2/vol19-2i.html (85 words)

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