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Gegenbauer   (Site not responding. Last check: 2007-10-14) After graduating from Berlin, Gegenbauer was appointed to a position at the University of Czernowitz (then in the Austrian Empire but now Chernovtsy, Ukraine) in 1875. After three years teaching in Innsbruck Gegenbauer was appointed full professor in 1881, then he was appointed full professor at the University of Vienna in 1893. Gegenbauer had many mathematical interests but was chiefly an algebraist. www-history.mcs.st-and.ac.uk /history/Mathematicians/Gegenbauer.html   (128 words)

Polynomials   (Site not responding. Last check: 2007-10-14) The monic part of a polynomial is the quotient of this polynomial by its leading coefficient. Legendre polynomials are a special case of Jacobi polynomials with the specific parameter values a = b = 0. The Gegenbauer polynomials are orthogonal with respect to the weight function (1-x^2)^(a-1/2) on the interval [-1,1]. www.xs4all.nl /~apinkus/refchapter6.html   (1885 words)

[No title]   (Site not responding. Last check: 2007-10-14) : Sure, your polynomial is nothing but a Gegenbauer polynomial of index +0.5 and it is well known that the derivative of any such Gegenbauer polynomial is another Gegenbauer polynomial (more precisely, everytime you derive, the degree is decreased by one and the index is incremented by one). What you are looking for is just the value at 1 of the Gegenbauer polynomial of degree n and index k+0.5 (i let you adjust the multiplicative factors which depend on the normalization choosen for the Gegenbauer polynomials). Last, but not least, since 1 is an endpoint of the interval which defines the orthogonality of the Gegenbauer polynomials, it appears that any such polynomial has there a value which can be easily expressed as a simple product (you just need the 2-terms recurrence involving " adjacent " Jacobi polynomials). www.math.niu.edu /~rusin/known-math/01_incoming/gegenbauer   (224 words)

An Integral Formula For Generalized Gegenbauer Polynomials And Jacobi Polynomials (ResearchIndex)   (Site not responding. Last check: 2007-10-14) Abstract: The generalized Gegenbauer polynomials are orthogonal polynomials with respect to the weight function jxj. An integral formula for these polynomials is proved, which serves as a transformation between h-harmonic polynomials associated with Z invariant weight functions on the plane. Polynomial Interpolation on the Unit Sphere and on the Unit Ball - Xu (2002) citeseer.ist.psu.edu /618165.html   (297 words)

Maxima Manual - Orthogonal Polynomials (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. The ultraspherical polynomials are also known as Gegenbauer polynomials. www.ma.utexas.edu /maxima/maxima_16.html   (1215 words)

Sankhya: The Indian Journal of Statistics   (Site not responding. Last check: 2007-10-14) A generalized Gegenbauer distribution with an additional parameter is defined in terms of generalized Gegenbauer polynomials. A number of relations associated with the polynomials are investigated. A probabilistic model for the distribution associated with the polynomials is indicated and a method of computing higher moments and cumulants is also given. sankhya.isical.ac.in /search/46b2/46b2018.html   (68 words)

The Mathematica Book Online: Advanced Mathematics in Mathematica | Mathematical Functions Gegenbauer polynomials GegenbauerC[n, m, x] can be viewed as generalizations of the Legendre polynomials to systems with Jacobi polynomials JacobiP[n, a, b, x] occur in studies of the rotation group, particularly in quantum mechanics. Legendre, Gegenbauer and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials. documents.wolfram.com /mathematica/book/section-3.2.10   (337 words)

GNU Scientific Library -- Reference Manual: Gegenbauer Functions   (Site not responding. Last check: 2007-10-14) The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultraspherical polynomials. These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3. This function computes an array of Gegenbauer polynomials C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda > -1/2, nmax >= 0. linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_115.html   (110 words)

orthpoly::gegenbauer -- the Gegenbauer (ultraspherical) polynomials   (Site not responding. Last check: 2007-10-14) a nonnegative integer: the degree of the polynomial. is an arithmetical expression, then the value of the Gegenbauer polynomial at this point is returned as an arithmetical expression. The Gegenbauer polynomials are given by the recursion formula www.sciface.com /STATIC/DOC25/de/orthpoly/gegenbauer.shtml   (202 words)

[No title]   (Site not responding. Last check: 2007-10-14) % % Special values: % % If ALPHA = 1, the Gegenbauer polynomials reduce to the Chebyshev % polynomials of the second kind. % Note that polynomials 0 through N will be computed. % % Input, real ALPHA, a parameter which is part of the definition of % the Gegenbauer polynomials. www.csit.fsu.edu /~burkardt/m_src/polpak/gegenbauer_poly.m   (105 words)

publices.htm Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegö class, preprint [math.CA/0311055], (2003). On the asymptotic expansion of the entropy of Gegenbauer polynomials, J. Asymptotics of Sobolev orthogonal polynomials for coherent pairs of measures, J. Approx. www.ual.es /GruposInv/Tapo/spanish/publices.htm   (2512 words)

POLPAK - Recursive Polynomials BP01 computes the values of the Bernstein polynomials at a point X. evaluates at X the Bernstein polynomials based in [A,B]. GEGENBAUER computes the Gegenbauer polynomials C(I,ALFA)(X) for I = 1 to N. computes the hail function. JACOBI evaluates the Jacobi polynomials at X. evaluates the Laguerre polynomials at X. evaluates the generalized Laguerre polynomials at X. evaluates the associated Laguerre polynomials Lnm at X. returns values of associated Legendre functions. orion.math.iastate.edu /burkardt/f_src/polpak/polpak.html   (438 words)

Functionals of Gegenbauer polynomials and D-dimensional hydrogenic momentum expectation values - Van Assche, nez, ... Abstract: The system of Gegenbauer polynomials fC n (x); n = 0; 1; : : : g is a classical family of polynomials orthogonal with respect to the weight function ! ...the entropy of the Gegenbauer polynomials of degree n and parameter ff. 3 The relativistic Hermite polynomial is a Gegenbauer polynomi.. citeseer.ist.psu.edu /assche99functional.html   (608 words)

POLPAK - Recursive Polynomials A variety of other polynomials and functions have been added. bpab.m, evaluates the Bernstein polynomials defined on the interval [A,B]; legendre_poly_coef.m, evaluates the coefficients of the Legendre polynomials P(N)(X); www.csit.fsu.edu /~burkardt/m_src/polpak/polpak.html   (828 words)

publicen.htm Non-diagonal Hermite-Sobolev orthogonal polynomials, Acta Applicandae Mathematicae 61, (1-3) (2000), 257-266. Asymptotic properties of Sobolev orthogonal polynomials, Invited lecture at VIII Simposium sobre Polinomios Ortogonales y Aplicaciones, Sevilla, September 22-26, 1997. Non-hermitian orthogonality and distribution of zeros of some classical polynomials with varying parameters, Short communication at the V International Symposium on Orthogonal Polynomials, Special Functions and Their Applications, Patras, Greece, September 20-24, 1999. www.ual.es /GruposInv/Tapo/english/publicen.htm   (2403 words)

Abstract of: Approximations of orthogonal polynomials in terms of Hermite polynomials   (Site not responding. Last check: 2007-10-14) Several orthogonal polynomials have limit forms in which Hermite polynomials show up. Examples are limits with respect to certain parameters of the Jacobi and Laguerre polynomials. In this paper we are interested in more details of these limits and we give asymptotic representations of several orthogonal polynomials in terms of Hermite polynomials. db.cwi.nl /rapporten/abstract.php?abstractnr=800   (155 words)

GNU Scientific Library -- Reference Manual - Gegenbauer Functions   (Site not responding. Last check: 2007-10-14) These functions evaluate the Gegenbauer polynomials @math{C^{(\lambda)}_n(x)} using explicit representations for @math{n =1, 2, 3}. These functions evaluate the Gegenbauer polynomial @c{$C^{(\lambda)}_n(x)$} @math{C^{(\lambda)}_n(x)} for a specific value of This function computes an array of Gegenbauer polynomials @math{C^{(\lambda)}_n(x)} for @math{n = 0, 1, 2, \dots, nmax}, subject to @math{\lambda > -1/2}, @c{$nmax \ge 0$} @math{nmax >= 0}. www.math.utah.edu /software/gsl/gsl-ref_115.html   (128 words)

\textsf{Method to compute the entropy of Gegenbauer polynomials with integer parameter. }   (Site not responding. Last check: 2007-10-14) Method to compute the entropy of Gegenbauer polynomials with integer parameter. We give a generalization of the method proposed by one of the authors [1], in order to develop an effective procedure for computing entropy for Gegenbauer polynomials, Information entropy of classical orthogonal polynomials and their application to the harmonic oscillator and Coulomb potentials. www.ugr.es /~alhambra2000/polyort/op11/op-011.html   (119 words)

Karl Dilcher's publications Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters, A generalization of Fibonacci polynomials and a representation of Gegenbauer polynomials of integer order, Polynomials related to expansions of certain rational functions in two variables, www.mscs.dal.ca /~dilcher/publications.html   (462 words)

Atlas: Application of Gegenbauer Polynomials and Mathieu Function to a Nonliner Dynamical System by N. K Choubey   (Site not responding. Last check: 2007-10-14) To avoid the difficulty of solving the nonlinear terms, we often neglect the nonlinear terms assuming them as they are small quantities but they must be taken into account to get the correct result. Denmann, Howard, Bhonsle and Garde have used Tchebycheff, Gegenbauer and Jacobi Polynomial approximation to study certain nonlinear system. 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 # came-23. atlas-conferences.com /cgi-bin/abstract/came-23   (240 words)

PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.), Vol. 62 (76), pp. 53--62, 1997   (Site not responding. Last check: 2007-10-14) Abstract: The principal object of this paper is to provide a natural further step toward the unified presentation of a class of Humbert's polynomials which generalizes the well known class of Gegenbauer, Legendre, Pincherle, Horadam, Kinney, Horadam-Pethe, Gould and Milovanovi\'c-Djordjevi\'c polynomials and many not so well-known polynomials. We shall give some basic relations involving the generalized Humbert polynomials and then take up several generating functions, hypergeometric representations and expansions in series of some relatively more familiar polynomials of Legendre, Gegenbauer, Hermite and Laguerre. Some of these results may be looked upon as providing useful extensions of the known results of Dilcher, Horadam, Sinha, Shreshtha and Milovanovi\'c-Djordjevi\'c. www.math.ethz.ch /EMIS/journals/PIMB/076/6.html   (156 words)

Information entropy of Gegenbauer polynomials and Gaussian quadrature For given values of n and l, this method requires the computation by means of recurrence relations of two auxiliary polynomials, P(x) and H(x), of degrees 2l - 2 and 2l - 4, respectively. Here it is shown that P(x) is related to the coefficients of the Gaussian quadrature formula for the Gegenbauer weights w In particular, reselling and systematic downloading of files is prohibited. stacks.iop.org /0305-4470/36/4857   (327 words)

Re: Gegenbauer polynomials mod 2? The volume on Orthogonal polynomials has a section on Gegenbauer Polynomials and Functions. "Lot-o-fun" <[EMAIL PROTECTED]> wrote in message news:[EMAIL PROTECTED] > I'm looking for any information about the Gegenbauer polynomials > (specifically, C_n^{(1)}(x)) in F_2[x] (F_2 = finite field with two > elements). I'll take anything I can get, but I'm particularly > interested in roots (in the algebraic closure of F_2), factors, etc. > I'd take any information about Legendre polys mod 2, since I might be > able to generalize. www.usenet.com /newsgroups/sci.math/msg23935.html   (125 words)

Kalnins (1990) Orthogonal polynomials on N-spheres: Gegenbauer, Jacobi, and Heun   (Site not responding. Last check: 2007-10-14) Kalnins (1990) Orthogonal polynomials on N-spheres: Gegenbauer, Jacobi, and Heun Orthogonal polynomials on N-spheres: Gegenbauer, Jacobi, and Heun To view the the latter's ratings, click on Chapters/Papers/Articles in the STATISTICS box, select a publication from the list that appears, and then click on either Quality or Interest in that publication's STATISTICS box. www.getcited.org /?PUB=103127134&showStat=Ratings   (83 words)

Professor Hari M. Srivastava Two-index multidimensional Gegenbauer polynomials and their integral representations A certain class of biorthogonal polynomials associated with the Laguerre polynomials Some polynomial expansions and generating functions associated with the Jacobi polynomials, International Conference on Boundary Value Problems, Special Functions and Fractional Calculus [Dedicated to the Ninetieth Birthday of Academician F. Gakhov (1906-1980)] held at Minsk, Belarus, February 16-20, 1996; in www.math.uvic.ca /faculty/harimsri   (4353 words)

Gegenbauer polynomials mod 2? I'm looking for any information about the Gegenbauer polynomials (specifically, C_n^{(1)}(x)) in F_2[x] (F_2 = finite field with two elements). I'll take anything I can get, but I'm particularly interested in roots (in the algebraic closure of F_2), factors, etc. I'd take any information about Legendre polys mod 2, since I might be able to generalize. Please check out one of the premium Usenet Newsgroup Service Providers below for access to Usenet. www.usenet.com /newsgroups/sci.math/msg23759.html   (72 words)

Math Preprint Series Title: Multiplicative renormalization and generating functions II Keywords: Orthogonal polynomials, Szego-Jacobi parameters, pre-generating functions, one-mode interating Fock space, Segal-Bargmann transform, Hermite polynomials, Charlier polynomials, Laguerre polynomials, Legendre polynomials, Gegenbauer polynomials, Chebyshev polynomials, negative binomial distributions Keywords: Gram-Schmidt orthogonalization process, orthogonal polynomials, Szego-Jacobi parameters, Fock space, Segal-Bargmann transform, Hermite polynomials, Charlier polynomials, Laguerre polynomials, Legendre polynomials, Gegenbauer polynomials, Chebyshev polynomials, Orthogonal polynomials associated with geometric distribution Keywords: fundamental group, polynomial cover, braid group, wiring diagram, intersection lattice www.math.lsu.edu /~preprint   (6092 words)

Special Functions - EqWorld Gegenbauer Polynomial, Eric W. Weisstein: from MathWorld—A Wolfram Web Resource Hermite Polynomial, Eric W. Weisstein: from MathWorld—A Wolfram Web Resource Laguerre Polynomial, Eric W. Weisstein: from MathWorld—A Wolfram Web Resource eqworld.ipmnet.ru /en/auxiliary/aux-specfunc.htm   (350 words)

Information entropy of Gegenbauer polynomials Received 6 March 2000, in final form 27 June 2000 The information entropy of Gegenbauer polynomials is relevant since this is related to the angular part of the information entropies of certain quantum mechanical systems such as the harmonic oscillator and the hydrogen atom in D dimensions. We give an effective method to compute the entropy for Gegenbauer polynomials with an integer parameter and obtain the first few terms in the asymptotic expansion as the degree of the polynomial tends to infinity. stacks.iop.org /0305-4470/33/6549   (262 words)

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