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Topic: Jacobi polynomials

	KARL JACOBI
		Jacobi was the son of a prosperous banker, and remained independently wealthy until the family fortune was lost in 1840.
		Jacobi was certainly not the first to use it; the "Jacobian" already appears in an 1815 paper of Cauchy.
		Jacobi was appointed to a position at the University of Königsberg in 1826.
	www.usna.edu /Users/math/meh/jacobi.html (597 words)

	Carl Gustav Jakob Jacobi Summary
		Jacobi's main area of interest was in the branch of mathematics that dealt with elliptic functions.
		Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the 2 square and 4 square theorems of Pierre de Fermat.
		It was in analytical development that Jacobi’s peculiar power mainly lay, and he made many important contributions of this kind to other departments of mathematics, as a glance at the long list of papers that were published by him in Crelle’s Journal and elsewhere from 1826 onwards will sufficiently indicate.
	www.bookrags.com /Carl_Gustav_Jakob_Jacobi (2337 words)

	Orthogonal polynomials - Wikipedia, the free encyclopedia
		In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p
		The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Stieltjes.
		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 (2018 words)

	Differential equations for generalized Laguerre and Jacobi polynomials
		We call these polynomials the generalized Jacobi polynomials, but sometimes they are also referred to as the Jacobi-type polynomials.
		This inversion method was found in a similar way as was done in [2] in the case of generalizations of the Charlier polynomials.
		In [13] it is shown that the Krall polynomials satisfy a sixth order differential equation of the form (1).
	aw.twi.tudelft.nl /~koekoek/diff (668 words)

	Publications of Yuan Xu
		Lecture notes on orthogonal polynomials of several variables, Summer School on Orthogonal Polynomials, Harmonic Analysis, Approximation and Applications, Inzell, Germany, Sept. 2001.
		Polynomial interpolation on the unit sphere and on the unit ball, Adv.
		Polynomial interpolation on the unit sphere, SIAM J. Numer.
	darkwing.uoregon.edu /~yuan/pub.html (2112 words)

	Springer Online Reference Works
		Wilson polynomials are closely related to classical orthogonal polynomials, since they are eigenfunctions of a second-order difference operator:
		Askey–Wilson polynomials are also orthogonal polynomial eigenfunctions of a second-order difference operator and they are believed to be the most general orthogonal polynomials with this property, in the sense that all other classes with this property can be obtained from them by specialization of parameters or as limit cases.
		The complete set of limit cases of Wilson and Racah polynomials is often written as a directed graph which is known as the Askey tableau, see the Appendix to [a2] as well as the references given there.
	eom.springer.de /w/w098000.htm (389 words)

	Legendre Polynomials
		The differential equation allows us to apply the polynomials to problems arising in mathematics and physics, among which is the important problem of the solution of Laplace's equation and spherical harmonics.
		The polynomials can also be found by solving the differential equation by determining the coefficients of a power series substituted in the equation.
		When differentiated n times, it becomes a polynomial of order n consisting of either all odd or all even powers of x, as n is odd or even.
	www.du.edu /~jcalvert/math/legendre.htm (1164 words)

	Maxima Manual - Orthogonal Polynomials
		[specfun package] return the Jacobi polynomial for integers n > -1 and a and b symbolic or a > -1 and b > -1.
		(The Jacobi polynomials are actually defined for all a and b ; however, the Jacobi polynomial weight (1-x)^a(1+x)^b isn't integrable for a <= -1 or b <= -1.
		Many functions in specfun are computed as a special case of the Jacobi polynomials; they also enjoy the speed boost from the modedeclared version of jacobi.
	www.ma.utexas.edu /maxima/maxima_16.html (1215 words)

	Generalizations of the umbral calculus
		The umbral calculus of [107] is restricted to the class of Sheffer polynomials.
		Viskov first extended the umbral calculus to so-called generalized Appell polynomials (or Boas-Buck polynomials) [129] and then went on to generalize this to arbitrary polynomials [130].
		The extension to generalized Appell polynomials makes it possible to apply umbral calculus to q-analysis [1,18,17,42,43,51,94] or important classes of orthogonal polynomials like the Jacobi polynomials [101].
	www.win.tue.nl /~sandro/hypersurvey/node10.html (505 words)

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

	[No title]
		{\it Perturbation of orthogonal polynomials on an arc of the unit circle, II\/} (with L.~Golinskii, F.~Pint\'er, and W.~Van~Assche), J.~Approx.
		{\it Perturbation of orthogonal polynomials on an arc of the unit circle\/} (with L.~Golinskii and W.~Van~Assche), J.~Approx.
		{\it Oscillatory behavior of orthogonal polynomials\/} (with A.~M\'at\'e and V.~Totik), Proc.
	www.math.ohio-state.edu /~nevai/MYMATH/mypubl_reverse.html (2755 words)

	Abstract of: Convergent asymptotic expansions of Charlier, Laguerre and Jacobi polynomials (Site not responding. Last check: 2007-10-11)
		Abstract of: Convergent asymptotic expansions of Charlier, Laguerre and Jacobi polynomials
		The expansions are given in terms of functions that are special cases of the given polynomials.
		The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.
	db.cwi.nl /rapporten/abstract.php?abstractnr=1423 (128 words)

	FOURIER SERIES IN ORTHOGONAL POLYNOMIALS (Site not responding. Last check: 2007-10-11)
		The main subject of the book is Fourier series in general orthogonal polynomials.
		The starting point of the technique in Chapters 4 and 5 is the representations of bilinear and trilinear forms obtained by the author.
		for experts in orthogonal polynomials, the most useful part of the book is the "Notes", where the author gives several precise additional comments on the results presented, on their origin and on their generalizations...
	www.worldscibooks.com /mathematics/4039.htm (296 words)

	Atlas: The behaviour of the complex zeros of the Laguerre and Jacobi polynomials. by Mark V. DeFazio (Site not responding. Last check: 2007-10-11)
		Atlas: The behaviour of the complex zeros of the Laguerre and Jacobi polynomials.
		The behaviour of the complex zeros of the Laguerre and Jacobi polynomials.
		Let {p\sb n(z)}\sbn=0\sp\infty be a sequence of polynomials of exact degree that satisfies the three term recurrence relation zp\sb n(z)=b\sb n p\sbn+1(z)+c\sb n p\sb n(z)+d\sb n p\sbn-1(z) with the initial conditions p\sb-1(z):=0 and p\sb 0(z)=1.
	atlas-conferences.com /cgi-bin/abstract/caku-59 (231 words)

	Graphical Discovery of a New Identity for Jacobi Polynomials -- from Mathematica Information Center
		During the summer of 1995 the authors engaged in an undergraduate research program that investigated various conjectures about orthogonal polynomials.
		While exploring the grahic capabilities of Mathematica, we generated Figure 1, which shows, on a single set of axes, the fifth-degree Jacobi polynomials, for beta = 0.9 and alpha taking the values 0, 1, 2,..., 6.
		We observed that the curves in the figure appear to intersect at extrema, but our advisors were skeptical about whether this actually happened.
	library.wolfram.com /infocenter/Articles/1780 (118 words)

	Springer Online Reference Works
		P.L. Chebyshev [1] proved that, among all polynomials of the form
		(among all polynomials (1)), and its norm is
		, there also exists a unique polynomial deviating least from zero; various properties of this polynomial are known (see [2], [5]).
	eom.springer.de /p/p073720.htm (203 words)

	On connections between Hankel, Laguerre and Jacobi transplantations, Krzysztof Stempak
		Proved are two results showing connections between the Hankel transplantation and a transplantation for a certain kind of Laguerre and Jacobi expansions.
		An asymptotic formula of Hilb's type for Laguerre and Jacobi polynomials is used.
		This is done by transferring a corresponding transplantation result for Jacobi expansions which was proved by Muckenhoupt.
	projecteuclid.org /getRecord?id=euclid.tmj/1113247646 (289 words)

	Zernike Polynoimials
		Zernike polynomials are often used since they are made up of terms that are of the same form as the types of aberrations often observed in optical tests.
		The radial polynomials are combined with sines and cosines rather than with a complex exponential.
		A useful feature of Zernike polynomials is that each term of the Zernikes minimizes the rms wavefront error to the order of that term.
	astron.berkeley.edu /~jrg/Aberrations/node11.html (456 words)

	Representation of Jacobi polynomials (Site not responding. Last check: 2007-10-11)
		A possible generalization to the case of Gegenbauer (ultraspherical) polynomials, i.e.
		Jacobi polynomials P_n^{(a,b)}(x) with a=b, has the form (1) (sin x)^{(2a)} P_n^{(a,a)}(cos x)=sum_{k=0}^infinity c_k sin((n+2k+1)x).
		The explicit values of c_k can be found by combining the following formulas in Abramowitz and Stegun: 8.7.1 (trigonometric expansion of Legendre function), 22.5.60 (ultraspherical polynomial expressed as Legendre function), 22.5.20 (Jacobi polynomial expressed as ultraspherical polynomial).
	cio.nist.gov /esd/emaildir/lists/opsftalk/msg00019.html (265 words)

	Documentation for Concepts 2.0
		The Jacobi version of the integration rule is used if this is non-zero.
		Computes the values of the derivatives of the Jacobi polynomials.
		Output: the zeros of the Jacobi polynomials in the elements 1,..., p.
	www.math.ethz.ch /~concepts/doxygen/html/quadratureBase_8hh.html (426 words)

	Affine Hecke Algebras and Orthogonal Polynomials - Cambridge University Press
		In recent years there has developed a satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters.
		These polynomials include as special cases: symmetric functions; zonal spherical functions on real and p-adic reductive Lie groups; the Jacobi polynomials of Heckman and Opdam; and the Askey-Wilson polynomials, which themselves include as special or limiting cases all the classical families of orthogonal polynomials in one variable.
		This first comprehensive and organised account of the subject aims to provide a unified foundation for this theory, to which the author has been a principal contributor.
	www.cambridge.org /catalogue/catalogue.asp?ISBN=0521824729 (267 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		The presentations will focus on group representations, free products of classical orthogonal polynomials, and possibly an introduction to random matrices.
		September 22, 2004: Everybody is invited to participate in the one-day workshop on reflection coefficients.
		For the Fall of 2003 the presentations will focus on the introduction of some techniques in positivity: Kolmogorov decompositions and the structure of positive definite kernels on the set of integers, and then on the discussion of orthogonal polynomials associated to various polynomial relations on noncommutative variables.
	www.utdallas.edu /~tiberiu/public_html/seminar.html (285 words)

	Biorthogonal polynomials suggested by the Jacobi polynomials., H. C. Madhekar, N. K. Thakare
		Biorthogonal polynomials suggested by the Jacobi polynomials., H. Madhekar, N. Thakare
		[8] Joseph D. E., Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math., 21 (1967), 303-314.
		[10] T. Prabhakar and N. Kashyap, A biorthogonal pair of polynomialsets suggested by a class of Jacobi polynomials, Indian J. Pure Appl.
	projecteuclid.org /getRecord?id=euclid.pjm/1102725253 (192 words)

	Home Page for Lance Littlejohn (Site not responding. Last check: 2007-10-11)
		On the right-definite and left-definite spectral theory of the Legendre polynomials, (with J. Arvesú and F. Marcellán), J.
		Jacobi-Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression, (with W. Everitt, R. Wellman and G. Yoon), submitted for publication.
		Orthogonal polynomials and differential operators I: the GKN theory, Invited lecture, Carlos III University, Madrid, Spain, March, 2002.
	www.math.usu.edu /~lance (2456 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		Approximation properties of mapped Jacobi polynomials and of interpolations based on mapped Jacobi-Gauss-Lobatto points are established.
		These results play an important role in numerical analysis of mapped Jacobi spectral methods.
		As examples of applications, optimal error estimates for several popular regular and singular mappings are derived.
	www.math.purdue.edu /~shen/pub/mapped_jacobi.txt (46 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.
		legendre_poly_coef.m, evaluates the coefficients of the Legendre polynomials P(N)(X);
	www.csit.fsu.edu /~burkardt/m_src/polpak/polpak.html (1019 words)

	Atlas: On the approximation of the Jacobi Polynomials by U. Elias (Site not responding. Last check: 2007-10-11)
		They are shown to converge faster then a geometric series, where the ratio of successive terms is
		We thus also demonstrate, that it is possible to approximate the Jacobi polynomials in the vicinity of x=1 as well as on the entire interval [1,
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqi-78.
	atlas-conferences.com /cgi-bin/abstract/caqi-78 (149 words)

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