32 FA15 Abstracts

Total Page:16

File Type:pdf, Size:1020Kb

32 FA15 Abstracts 32 FA15 Abstracts IP1 metric simple exclusion process and the KPZ equation. In Vector-Valued Nonsymmetric and Symmetric Jack addition, the experiments of Takeuchi and Sano will be and Macdonald Polynomials briefly discussed. For each partition τ of N there are irreducible representa- Craig A. Tracy tions of the symmetric group SN and the associated Hecke University of California, Davis algebra HN (q) on a real vector space Vτ whose basis is [email protected] indexed by the set of reverse standard Young tableaux of shape τ. The talk concerns orthogonal bases of Vτ -valued polynomials of x ∈ RN . The bases consist of polyno- IP6 mials which are simultaneous eigenfunctions of commuta- Limits of Orthogonal Polynomials and Contrac- tive algebras of differential-difference operators, which are tions of Lie Algebras parametrized by κ and (q, t) respectively. These polynomi- als reduce to the ordinary Jack and Macdonald polynomials In this talk, I will discuss the connection between superin- when the partition has just one part (N). The polynomi- tegrable systems and classical systems of orthogonal poly- als are constructed by means of the Yang-Baxter graph. nomials in particular in the expansion coefficients between There is a natural bilinear form, which is positive-definite separable coordinate systems, related to representations of for certain ranges of parameter values depending on τ,and the (quadratic) symmetry algebras. This connection al- there are integral kernels related to the bilinear form for lows us to extend the Askey scheme of classical orthogonal the group case, of Gaussian and of torus type. The mate- polynomials and the limiting processes within the scheme. rial on Yang-Baxter graphs and Macdonald polynomials is In particular, for superintegrable systems in 3D, the poly- based on joint work with J.-G. Luque. nomial representations of quadratic algebras are given in terms of two-variable polynomials and the two-variable Charles F. Dunkl analog of the Askey scheme, including the quadratic Racah University of Virginia algebra, will be discussed along with the limiting processes [email protected] within the scheme. Sarah Post IP2 University of Hawaii at Manoa Two Variable q-Polynomials Department of Mathematics [email protected] Abstract not available at this time. Mourad Ismail IP7 University of Central Florida A New Look at Classical Orthogonal Polynomials [email protected] There are two possible definitions of classical orthogo- nal polynomials:(i) they satisfy a second order differential IP3 or difference equation;(ii) (generalized) derivative of them On the Asymptotic Behavior of a Log Gas in the gives again orthogonal polynomials. Both definitions are Bulk Scaling Limit in the Presence of a Varying related with concrete forms of corresponding operators. We External Potential propose a new approach dealing with some abstract um- bral operators. This gives a wide generalization of a notion Abstract not available at time of publication. of classical orthogonal polynomials. Percy Deift Alexei Zhedanov Courant Institute Donetsk Institute for Physics and Engineering, Ukraine New York University [email protected] [email protected] IP4 IP8 Title Not Available at Time of Publication Asymptotic and Numerical Aspects of Special Functions Abstract not available at this time. For the numerical evaluation of special functions, asymp- Olga Holtz totic expansions are an important tool. The standard ex- University of California, Berkeley pansions can be used rather straightforwardly. The so- Technische Universitaet Berlin called uniform expansions need more attention, especially [email protected] for critical values of secondary parameters in the asymp- totic problem. For example, the Airy-type expansion of the Bessel function Jν (z) can be used for large domains of the IP5 argument and order, but for the transition value z = ν spe- Integrable Probability and the Role of Painlev cial methods are needed for computing the coefficients. We Functions mention several methods for handling this type of problem. We start with a few examples for which Maple and Mathe- We will review various models in probability that are inte- matica have problems in the evaluation of well-known spe- grable in the sense that various distribution functions can cial functions, like the Kummer U-function, for medium- be explicitly evaluated in terms of Painleve functions and sized values of the parameters. We discuss recent activities their generalizations. We develop in more detail a class in the Santander-Amsterdam project on the evaluation of of stochastic growth models that belong to the Kardar- special functions, in particular for certain cumulative dis- Parisis-Zhang (KPZ) uni- versality class such as the asym- tribution functions. We start with the incomplete gamma FA15 Abstracts 33 functions, and we give recent results for the non central chi- polynomials, which are multivariate orthogonal polynomi- squared or the non central gamma distribution, also called als. Recently Eric Rains defined moments of Koornwinder Marcum Q−function in radar detection and communica- polynomials at q=t, which appear to be polynomials with tion problems. This is joint work with Amparo Gil and positive coefficients when written appropriately in the pa- Javier Segura (University of Cantabria, Santander, Spain). rameters of the ASEP. I’ll explain joint work with Sylvie Corteel in which we show that Koornwinder moments at q=t are related to the 2-species ASEP, an exclusion process Nico M. Temme involving two different types of particles. I’ll also describe Centrum Wiskunde & Informatica complementary work of Olga Mandelshtam and Xavier Vi- The Netherlands ennot providing a combinatorial description of the station- [email protected] ary distribution of the 2-species ASEP. Lauren Williams IP9 University of California Berkeley Multivariate Orthogonal Polynomials and Modified [email protected] Moment Functionals Multivariate orthogonal polynomials can be defined by SP1 d means a measure defined on a domain on R . A very impor- SIAG/OPSF Gbor Szeg Prize Announcement and tant class of multivariate orthogonal polynomials is called Lecture classical because the measure satisfies a matrix analogue of the Pearson differential equation as well as the orthogo- Abstract not available at time of publication.. nal polynomials are the eigenfunctions of a partial second order differential operator. In this talk, we present old and Karl Liechty new results on classical multivariate orthogonal polynomi- DePaul University als. In particular, some classical multivariate orthogonal Department of Mathematical Sciences polynomials and some useful modifications will be studied, [email protected] as well as their impact into the useful properties of the orthogonal polynomials. We study the so–called Uvarov modification obtained by adding to the measure one or a CP1 finite set of mass points. Recently, Christoffel modifica- Killip-Simon Problem and Jacobi Flow on Gsmp tion in several variables, that is, the modification obtained Matrices by multiplying the measure times a polynomial, has been studied in the frame of linear relations. D. Damanik, R. Killip and B. Simon completely described the spectral properties of Jacobi matrices J+,whicharein 2 Teresa E. P´erez asence perturbations of the isospectral torus of periodic Universidad de Granada Jacobi matrices. The spectrum of a periodic Jacobi matrix Departamento de Matem´atica Aplicada is a system of intervals of a very specific nature. Jointly [email protected] with P. Yuditskii, we generalize this result to spectral sets, which are arbitrary finite system of intervals. IP10 Benjamin Eichinger Hypergeometric Series: On Number Theory’s Se- Institute of Analysis, Johannes Kepler University Linz cret Service [email protected] A natural outcome of the theory of generalized hypergeo- CP1 metric functions are rational approximations to the values of Riemann’s zeta functions and alike mathematical con- A Matrix Approach for the Semiclassical and Co- stants. In my talk I plan to outline the way it goes (hyper- herent Orthogonal Polynomials geometric series, hypergeometric Barnes- and Euler-type integrals) and stress on some recent achievements — the We obtain a matrix characterization of semiclassical or- current best irrationality measures of π (due to Salikhov), thogonal polynomials in terms of the Jacobi matrix as- of log 2 (due to Marcovecchio) and of ζ(2). The final part sociated with the multiplication operator in the basis of of the talk will discuss some linear and algebraic indepen- orthogonal polynomials, and the lower triangular matrix dence results that make use of generalized hypergeometric that represents the orthogonal polynomials in terms of the functions. monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals. Wadim Zudilin University of Newcastle Lino G. Garza [email protected] Universidad Carlos III de Madrid [email protected] IP11 Francisco Marcell´an Orthogonal Polynomials and the 2-Species ASEP Universidad Carlos III de Madrid Instituto de Ciencias Matem´aticas (ICMAT) The asymmetric exclusion process (ASEP) is a model of [email protected] particles hopping on a 1-dimensional lattice with open boundaries. The partition function of this model is related Luis E. Garza to moments of Askey-Wilson polynomials. Askey-Wilson Departamento Matematicas,Universidad Nacional polynomials are at the top of the hierarchy of orthogonal Colombia polynomials, and are also
Recommended publications
  • Math/Library Special Functions Quick Tips on How to Use This Online Manual
    Fortran Subroutines for Mathematical Applications Math/Library Special Functions Quick Tips on How to Use this Online Manual Click to display only the page. Click to go back to the previous page from which you jumped. Click to display both bookmark and the page. Click to go to the next page. Double-click to jump to a topic Click to go to the last page. when the bookmarks are displayed. Click to jump to a topic when the Click to go back to the previous view and bookmarks are displayed. page from which you jumped. Click to display both thumbnails Click to return to the next view. and the page. Click and use to drag the page in vertical Click to view the page at 100% zoom. direction and to select items on the page. Click and drag to page to magnify Click to fit the entire page within the the view. window. Click and drag to page to reduce the view. Click to fit the page width inside the window. Click and drag to the page to select text. Click to find part of a word, a complete word, or multiple words in a active document. Click to go to the first page. Printing an online file: Select Print from the File menu to print an online file. The dialog box that opens allows you to print full text, range of pages, or selection. Important Note: The last blank page of each chapter (appearing in the hard copy documentation) has been deleted from the on-line documentation causing a skip in page numbering before the first page of the next chapter, for instance, Chapter 1 in the on-line documentation ends on page 317 and Chapter 2 begins on page 319.
    [Show full text]
  • IMSL Fortran Library User's Guide MATH/LIBRARY Special Functions
    IMSL Fortran Library User’s Guide MATH/LIBRARY Special Functions Mathematical Functions in Fortran Trusted For Over 30 Years IMSL MATH/LIBRARY User’s Guide Special Functions Mathematical Functions in Fortran P/N 7681 [ www.vni.com ] Visual Numerics, Inc. – United States Visual Numerics International Ltd. Visual Numerics SARL Corporate Headquarters Centennial Court Immeuble le Wilson 1 2000 Crow Canyon Place, Suite 270 Easthampstead Road 70, avenue due General de Gaulle San Ramon, CA 94583 Bracknell, Berkshire RG12 1YQ F-92058 PARIS LA DEFENSE, Cedex PHONE: 925-807-0138 UNITED KINGDOM FRANCE FAX: 925-807-0145 e-mail: [email protected] PHONE: +44 (0) 1344-458700 PHONE: +33-1-46-93-94-20 Westminster, CO FAX: +44 (0) 1344-458748 FAX: +33-1-46-93-94-39 PHONE: 303-379-3040 e-mail: [email protected] e-mail: [email protected] Houston, TX PHONE: 713-784-3131 Visual Numerics S. A. de C. V. Visual Numerics International GmbH Visual Numerics Japan, Inc. Florencia 57 Piso 10-01 Zettachring 10 GOBANCHO HIKARI BLDG. 4TH Floor Col. Juarez D-70567Stuttgart 14 GOBAN-CHO CHIYODA-KU Mexico D. F. C. P. 06600 GERMANY TOKYO, JAPAN 102 MEXICO PHONE: +49-711-13287-0 PHONE: +81-3-5211-7760 PHONE: +52-55-514-9730 or 9628 FAX: +49-711-13287-99 FAX: +81-3-5211-7769 FAX: +52-55-514-4873 e-mail: [email protected] e-mail: [email protected] Visual Numerics, Inc. Visual Numerics Korea, Inc. World Wide Web site: http://www.vni.com 7/F, #510, Sect. 5 HANSHIN BLDG.
    [Show full text]
  • Fortran Math Special Functions Library
    IMSL® Fortran Math Special Functions Library Version 2021.0 Copyright 1970-2021 Rogue Wave Software, Inc., a Perforce company. Visual Numerics, IMSL, and PV-WAVE are registered trademarks of Rogue Wave Software, Inc., a Perforce company. IMPORTANT NOTICE: Information contained in this documentation is subject to change without notice. Use of this docu- ment is subject to the terms and conditions of a Rogue Wave Software License Agreement, including, without limitation, the Limited Warranty and Limitation of Liability. ACKNOWLEDGMENTS Use of the Documentation and implementation of any of its processes or techniques are the sole responsibility of the client, and Perforce Soft- ware, Inc., assumes no responsibility and will not be liable for any errors, omissions, damage, or loss that might result from any use or misuse of the Documentation PERFORCE SOFTWARE, INC. MAKES NO REPRESENTATION ABOUT THE SUITABILITY OF THE DOCUMENTATION. THE DOCU- MENTATION IS PROVIDED "AS IS" WITHOUT WARRANTY OF ANY KIND. PERFORCE SOFTWARE, INC. HEREBY DISCLAIMS ALL WARRANTIES AND CONDITIONS WITH REGARD TO THE DOCUMENTATION, WHETHER EXPRESS, IMPLIED, STATUTORY, OR OTHERWISE, INCLUDING WITHOUT LIMITATION ANY IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PAR- TICULAR PURPOSE, OR NONINFRINGEMENT. IN NO EVENT SHALL PERFORCE SOFTWARE, INC. BE LIABLE, WHETHER IN CONTRACT, TORT, OR OTHERWISE, FOR ANY SPECIAL, CONSEQUENTIAL, INDIRECT, PUNITIVE, OR EXEMPLARY DAMAGES IN CONNECTION WITH THE USE OF THE DOCUMENTATION. The Documentation is subject to change at any time without notice. IMSL https://www.imsl.com/ Contents Introduction The IMSL Fortran Numerical Libraries . 1 Getting Started . 2 Finding the Right Routine . 3 Organization of the Documentation . 4 Naming Conventions .
    [Show full text]
  • Orthogonal Functions: the Legendre, Laguerre, and Hermite Polynomials
    ORTHOGONAL FUNCTIONS: THE LEGENDRE, LAGUERRE, AND HERMITE POLYNOMIALS THOMAS COVERSON, SAVARNIK DIXIT, ALYSHA HARBOUR, AND TYLER OTTO Abstract. The Legendre, Laguerre, and Hermite equations are all homogeneous second order Sturm-Liouville equations. Using the Sturm-Liouville Theory we will be able to show that polynomial solutions to these equations are orthogonal. In a more general context, finding that these solutions are orthogonal allows us to write a function as a Fourier series with respect to these solutions. 1. Introduction The Legendre, Laguerre, and Hermite equations have many real world practical uses which we will not discuss here. We will only focus on the methods of solution and use in a mathematical sense. In solving these equations explicit solutions cannot be found. That is solutions in in terms of elementary functions cannot be found. In many cases it is easier to find a numerical or series solution. There is a generalized Fourier series theory which allows one to write a function f(x) as a linear combination of an orthogonal system of functions φ1(x),φ2(x),...,φn(x),... on [a; b]. The series produced is called the Fourier series with respect to the orthogonal system. While the R b a f(x)φn(x)dx coefficients ,which can be determined by the formula cn = R b 2 , a φn(x)dx are called the Fourier coefficients with respect to the orthogonal system. We are concerned only with showing that the Legendre, Laguerre, and Hermite polynomial solutions are orthogonal and can thus be used to form a Fourier series. In order to proceed we must define an inner product and define what it means for a linear operator to be self- adjoint.
    [Show full text]
  • Generalizations and Specializations of Generating Functions for Jacobi, Gegenbauer, Chebyshev and Legendre Polynomials with Definite Integrals
    Journal of Classical Analysis Volume 3, Number 1 (2013), 17–33 doi:10.7153/jca-03-02 GENERALIZATIONS AND SPECIALIZATIONS OF GENERATING FUNCTIONS FOR JACOBI, GEGENBAUER, CHEBYSHEV AND LEGENDRE POLYNOMIALS WITH DEFINITE INTEGRALS HOWARD S. COHL AND CONNOR MACKENZIE Abstract. In this paper we generalize and specialize generating functions for classical orthogo- nal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generat- ing functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given. 1. Introduction This paper concerns itself with analysis of generating functions for Jacobi, Gegen- bauer, Chebyshev and Legendre polynomials involving generalization and specializa- tion by re-expression of Gauss hypergeometric generating functions for these orthog- onal polynomials. The generalizations that we present here are for two of the most important generating functions for Jacobi polynomials, namely [4, (4.3.1–2)].1 In fact, these are the first two generating functions which appear in Section 4.3 of [4]. As we will show, these two generating functions, traditionally expressed in terms of Gauss hy- pergeometric functions, can be re-expressed in terms of associated Legendre functions (and also in terms of Ferrers functions, associated Legendre functions on the real seg- ment ( 1,1)). Our Jacobi polynomial generating function generalizations, Theorem 1, Corollary− 1 and Corollary 2, generalize the generating function for Gegenbauer polyno- mials.
    [Show full text]
  • Arxiv:2008.08079V2 [Math.FA] 29 Dec 2020 Hypergroups Is Not Required)
    HARMONIC ANALYSIS OF LITTLE q-LEGENDRE POLYNOMIALS STEFAN KAHLER Abstract. Many classes of orthogonal polynomials satisfy a specific linearization prop- erty giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to harmonic and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as L1-algebras, associated with underlying orthogonal polynomials or with the corresponding orthogonalization measures. The individual be- havior strongly depends on these underlying polynomials. We study the little q-Legendre polynomials, which are orthogonal with respect to a discrete measure. Their L1-algebras have been known to be not amenable but to satisfy some weaker properties like right character amenability. We will show that the L1-algebras associated with the little q- Legendre polynomials share the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these L1-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner deriva- tion), which is known to be shared by any L1-algebra of a locally compact group. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on continued fractions, character estimations and asymptotic behavior. 1. Introduction 1.1. Motivation. One of the most famous results of mathematics, the ‘Banach–Tarski paradox’, states that any ball in d ≥ 3 dimensions can be split into a finite number of pieces in such a way that these pieces can be reassembled into two balls of the original size. It is also well-known that there is no analogue for d 2 f1; 2g, and the Banach–Tarski paradox heavily relies on the axiom of choice [37].
    [Show full text]
  • Shifted Jacobi Polynomials and Delannoy Numbers
    SHIFTED JACOBI POLYNOMIALS AND DELANNOY NUMBERS GABOR´ HETYEI A` la m´emoire de Pierre Leroux Abstract. We express a weigthed generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago [8], [13], [14], to all Delannoy numbers and certain Jacobi polynomials. Another specializa- tion provides a weighted lattice path enumeration model for shifted Jacobi polynomials and allows the presentation of a combinatorial, non-inductive proof of the orthogonality of Jacobi polynomials with natural number parameters. The proof relates the orthogo- nality of these polynomials to the orthogonality of (generalized) Laguerre polynomials, as they arise in the theory of rook polynomials. We also find finite orthogonal polynomial sequences of Jacobi polynomials with negative integer parameters and expressions for a weighted generalization of the Schr¨odernumbers in terms of the Jacobi polynomials. Introduction It has been noted more than fifty years ago [8], [13], [14] that the diagonal entries of the Delannoy array (dm,n), introduced by Henri Delannoy [5], and the Legendre polynomials Pn(x) satisfy the equality (1) dn,n = Pn(3), but this relation was mostly considered a “coincidence”. An important observation of our present work is that (1) can be extended to (α,0) (2) dn+α,n = Pn (3) for all α ∈ Z such that α ≥ −n, (α,0) where Pn (x) is the Jacobi polynomial with parameters (α, 0). This observation in itself is a strong indication that the interaction between Jacobi polynomials (generalizing Le- gendre polynomials) and the Delannoy numbers is more than a mere coincidence.
    [Show full text]
  • Algorithms for Classical Orthogonal Polynomials
    Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustr. 7, D-14195 Berlin - Dahlem Wolfram Ko epf Dieter Schmersau Algorithms for Classical Orthogonal Polynomials at Berlin Fachb ereich Mathematik und Informatik der Freien Universit Preprint SC Septemb er Algorithms for Classical Orthogonal Polynomials Wolfram Ko epf Dieter Schmersau koepfzibde Abstract In this article explicit formulas for the recurrence equation p x A x B p x C p x n+1 n n n n n1 and the derivative rules 0 x p x p x p x p x n n+1 n n n n1 n and 0 p x p x x p x x n n n n1 n n resp ectively which are valid for the orthogonal p olynomial solutions p x of the dierential n equation 00 0 x y x x y x y x n of hyp ergeometric typ e are develop ed that dep end only on the co ecients x and x which themselves are p olynomials wrt x of degrees not larger than and resp ectively Partial solutions of this problem had b een previously published by Tricomi and recently by Yanez Dehesa and Nikiforov Our formulas yield an algorithm with which it can b e decided whether a given holonomic recur rence equation ie one with p olynomial co ecients generates a family of classical orthogonal p olynomials and returns the corresp onding data density function interval including the stan dardization data in the armative case In a similar way explicit formulas for the co ecients of the recurrence equation and the dierence rule x rp x p x p x p x n n n+1 n n n n1 of the classical orthogonal p olynomials of a discrete variable are given that dep end only
    [Show full text]
  • A Uniqueness Theorem for the Legendre and Hermite
    A UNIQUENESSTHEOREM FOR THELEGENDRE ANDHERMITE POLYNOMIALS* BY K. P. WILLIAMS 1. If we replace y in the expansion of (l-\-y)~v by 2xz-\-z2, the coefficient of zn will, when x is replaced by —x, be the generalized polynomial L'ñ(x) of Legendre. It is also easy to show that the Hermitian polynomial Hn(x), usually defined by is the coefficient of 2"/«! in the series obtained on replacing y in the expansion of e~y by the same expression 2xz-\-z2. Furthermore, there is a simple recursion formula between three successive Legrendre polynomials and between three successive Hermitian polynomials. These facts suggest the following problem. Let a> ftg <p(y) = «o + «i 2/ +y,-2/2+3y ¡y3+ ••• and put (p(2xz 4z2) = P„ + Px (x) t + Ps(x) z2+---. To what extent is the generating function q>(y) determined if it is known that a simple recursion relation exists between three of the successive polynomials P0, Pi(x), P%(x), •••? We shall find that the generalized Legendre polynomials and those of Hermite possess a certain uniqueness in this regard. 2. We have PÁX) = «! 7i«íp(2!rí + £,).._0- When we make use of the formula for the «th derivative of a function of a function given by Faà de Bruno,t we find without difficulty P»(x) = Zjtt^xy, 'Presented to the Society, October 25, 1924. "¡■Quarterly Journal of Mathematics, vol. 1, p. 359. 441 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 442 K- p- WILLIAMS [October where the summation extends to all values of i and j subject to the relation i + 2j = n.
    [Show full text]
  • ME201/MTH281/ME400/CHE400 Legendre Polynomials
    ME201/MTH281/ME400/CHE400 Legendre Polynomials 1. Introduction This notebook has three objectives: (1) to summarize some useful information about Legendre polynomials, (2) to show how to use Mathematica in calculations with Legendre polynomials, and (3) to present some examples of the use of Legendre polynomials in the solution of Laplace's equation in spherical coordinates. In our course, the Legendre polynomials arose from separation of variables for the Laplace equation in spherical coordinates, so we begin there. The basic spherical coordinate system is shown below. The location of a point P is specified by the distance r of the point from the origin, the angle f between the position vector and the z-axis, and the angle q from the x-axis to the projection of the position vector onto the xy plane. The Laplace equation for a function F(r, f, q) is given by 1 ¶ ¶F 1 ¶ ¶F 1 ¶2 F “2 F = r2 + sinf + = 0 . (1) r2 ¶r ¶r r2 sinf ¶f ¶f r2 sin2 f ¶q2 In this notebook, we will consider only axisymmetric solutions of (1) -- that is, solutions which depend on r and f but not on q. Then equation (1) reduces to 1 ¶ ¶F 1 ¶ ¶F r2 + sinf = 0 . (2) r2 ¶r ¶r r2 sinf ¶f ¶f As we showed in class by a rather lengthy analysis, equation (2) has separated solutions of the form 2 legendre.nb n -Hn+1L r Pn HcosfL and r Pn HcosfL , (3) where n is a non-negative integer and Pn is the nth Legendre polynomial.
    [Show full text]
  • Orthogonal Polynomials and Classical Orthogonal Polynomials
    International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 10, October 2018, pp. 1613–1630, Article ID: IJMET_09_10_164 Available online at http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed ORTHOGONAL POLYNOMIALS AND CLASSICAL ORTHOGONAL POLYNOMIALS DUNIA ALAWAI JARWAN Education for Girls College, Al-Anbar University, Ministry of Higher Education and Scientific Research, Iraq ABSTRACT The focus of this project is to clarify the concept of orthogonal polynomials in the case of continuous internal and discrete points on R and the Gram – Schmidt orthogonalization process of conversion to many orthogonal limits and the characteristics of this method. We have highlighted the classical orthogonal polynomials as an example of orthogonal polynomials because of they are great importance in physical practical applications. In this project, we present 3 types (Hermite – Laguerre – Jacobi) of classical orthogonal polynomials by clarifying the different formulas of each type and how to reach some formulas, especially the form of the orthogonality relation of each. Keywords: Polynomials, Classical Orthogonal, Monic Polynomial, Gram – Schmidt Cite this Article Dunia Alawai Jarwan, Orthogonal Polynomials and Classical Orthogonal Polynomials, International Journal of Mechanical Engineering and Technology, 9(10), 2018, pp. 1613–1630. http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 1. INTRODUCTION The mathematics is the branch where the lots of concepts are included. An orthogonality is the one of the concept among them. Here we focuse on the orthogonal polynomial sequence. The orthogonal polynomial are divided in two classes i.e. classical orthogonal polynomials, Discrete orthogonal polynomials and Sieved orthogonal polynomials .There are different types of classical orthogonal polynomials such that Jacobi polynomials, Associated Laguerre polynomials and Hermite polynomials.
    [Show full text]
  • Orthogonal Polynomials: an Illustrated Guide
    Orthogonal Polynomials: An Illustrated Guide Avaneesh Narla December 10, 2018 Contents 1 Definitions 1 2 Example 1 2 3 Three-term Recurrence Relation 3 4 Christoffel-Darboux Formula 5 5 Zeros 6 6 Gauss Quadrature 8 6.1 Lagrange Interpolation . .8 6.2 Gauss quadrature formula . .8 7 Classical Orthogonal Polynomials 11 7.1 Hermite Polynomials . 11 7.2 Laguerre Polynomials . 12 7.3 Legendre Polynomials . 14 7.4 Jacobi Polynomials . 16 7.5 Chebyshev Polynomials of the First Kind . 17 7.6 Chebyshev Polynomials of the Second Kind . 19 7.7 Gegenbauer polynomials . 20 1 Definitions Orthogonal polynomials are orthogonal with respect to a certain function, known as the weight function w(x), and a defined interval. The weight function must be continuous and positive such that its moments (µn) exist. Z b n µn := w(x)x dx; n = 0; 1; 2; ::: a The interval may be infinite. We now define the inner product of two polynomials as follows Z 1 hf; giw(x) := w(x)f(x)g(x) dx −∞ 1 We will drop the subscript indicating the weight function in future cases. Thus, as always, a 1 sequence of polynomials fpn(x)gn=0 with deg(pn(x)) = n are called orthogonal polynomials for a weight function w if hpm; pni = hnδmn Above, the delta function is the Kronecker Delta Function There are two possible normalisations: If hn = 1 8n 2 f0; 1; 2:::g, the sequence is orthonormal. If the coefficient of highest degree term is 1 for all elements in the sequence, the sequence is monic.
    [Show full text]