DOCSLIB.ORG

A Probabilistic Counterpart of Askey Scheme for Continuous Polynomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Probabilistic Counterpart of Askey Scheme for Continuous Polynomials (), 1–27 /.. A probabilistic counterpart of Askey scheme for continuous polynomials Jean-Renaud Pycke Abstract. Random variables corresponding to weight functions of the Askey scheme for continuous polynomials are introduced. Beside normal, Beta and Gamma distributions four new families of real random variables corresponding to Wilson, continuous dual Hahn, continuous Hahn and Meixner-Pollaczek polynomials are discussed. Formulas for the moments and asymptotic behavior of the tails of the distributions are provided. Keywords. Askey scheme, orthogonal polynomials, parametric model. 2010 Mathematics Subject Classification. 42C05, 60E05. 1 Introduction The main goal of this paper is to introduce families of real random variables de- rived in a natural way from the orthogonal families of the Askey-scheme for con- tinuous polynomials. Both graphs (1.9) and (1.10) can be seen as probabilistic counterparts of the Askey scheme for continuous polynomials (1.1). The Askey scheme for continuous and discrete polynomials represents an out- standing achievement in the theory of special functions over the last decades. The Askey-scheme for continuous polynomials encompasses the theory of classical orthogonal polynomials of Laguerre, Hermite and Jacobi. It provides relations between these celebrated sequences of polynomials and includes them in a much wider class. The latter is much less familiar to non specialists and contains Wat- son, continuous dual Hahn, continuous Hahn, Meixner-Pollaczek polynomials. Whereas the connection between classical polynomials and parametric families of random variables as Gamma, Beta or Gaussian distributions are well-known, the random variables corresponding to this wider family have not been studied in a systematic way, even in a comprehensive survey of various relationships between orthogonal polynomials and probability theory as [8]. Excepting the Meixner- Pollaczek distribution, see e.g. [7] p.154, the use of these distributions as models remains a widely open field. Therefore the aim of the present paper is to con- tribute to filling this gap and provide basic results for further use of the fruitful Askey scheme in the field of probability and statistics. 2 J.-R. Pycke Concerning the Askey scheme, notations, definitions and results about continu- ous polynomials of the Askey scheme we deal with in the present paper, complete and self-contained references for all formulas are [3], [4]. See also Chapter 1 and Appendix E in [8] and [5]. The Askey scheme of continuous polynomials admits the graph representation Wilson (1.1) r w Continuous dual Hahn Continuous Hahn t ' Meixner-Pollaczek Jacobi ) w Laguerre $ ~ Hermite Each vertex corresponds to a parametric family of sequences of orthogonal polyno- mials named after the mathematician who introduced or studied the corresponding family. The arrows represent mathematical relations that will be made explicit below. With polynomial notations the graph becomes 2 Wk(x ; a; b; c; d) s w 2 (a;b) Sk(x ; a; b; c) Hk (x) x ' P (λ)(x; φ) P (α,β)(x) & w L(α)(x) Hk(x) Probabilistic Askey scheme 3 Arrows correspond to the limit relations 2 Wk(x ; a; b; c; d) 2 lim = Sk(x ; a; b; c) d!1 (a + d)k 2 Wk((x + t) ; a − it; b − it; a + it; b + it) lim = Hk(x; a; b) t!1 ((−2t)kk! 2 1−x α+1 α+1 β+1 β+1 Wk(t ; ; ; + it; − it) lim 2 2 2 2 2 = P (α,β)(x) t!1 k!t2k S ((x − t)2; λ + it; λ − it; t cot φ) lim k = P (λ)(x; φ) t!1 t k k!( sin φ )k Hλ+it;−t tan φ(x − t) lim k = P (λ)(x; φ) t!1 k it k i ( cos φ )k α+1+it ; β+1−it H 2 2 (− xt ) lim k 2 = P (α,β)(x) t!1 (−t)k k α+1 ( 2 ) x α lim Pk (− ; φ) = Lk (x) φ!0 2φ p x λ − λ cos φ H (x) lim λ−k=2P (λ)( ; φ) = k λ!1 k sin φ k! 2x lim P (α,β)(1 − ) = L(α)(x) β!1 k β k x H (x) lim α−k=2P (α,α)(p ) = k α!1 k α 2kk! k (α) p (−1) lim L (x 2α + α) = Hk(x): α!1 k k! Each family of orthogonal polynomials is related to a real weight-function via orthogonality relations (2.1). In view of applications in probability theory and statistics we will limit ourselves to parameters for which the weight function is non-negative. In that case after normalization the weight-function gives rise to a probability density function (p.d.f). The weight functions of the Askey-Wilson scheme and hypotheses on coefficients are the following, the name of the corre- 4 J.-R. Pycke sponding orthogonal families being given each case. 2 1 G(a + ix)G(b + ix)G(c + ix)G(d + ix) fa;b;c;d(x) = (Wilson) 2π G(2ix) x > 0; R(a; b; c; d) > 0; non-real parameters occuring in conjugate pairs; (1.2) 2 1 G(a + ix)G(b + ix)G(c + ix) fa;b;c(x) = (continuous dual Hahn) 2π G(2ix) x > 0; R(a; b; c) > 0; non-real parameters occurring in a conjugate pair; (1.3) −1 2 fa;b(x) = (2π) jG(a + ix)G(b + ix)j ; (continuous Hahn) x 2 R; R(a; b) > 0; (1.4) f (λ)(x; φ) = (2π)−1e(2φ−π)xjG(λ + ix)j2 (Meixner-Pollaczek) x 2 R; λ > 0; 0 < φ < π; (1.5) f (α,β)(x) = (1 − x)α(1 + x)β; −1 < x < 1; α; β > −1 (Jacobi); (1.6) f (α)(x) = xαe−x; x > 0; α > −1 (Laguerre) ; (1.7) −x2 f0(x) = e ; x 2 R (Hermite). (1.8) Remark 1.1. Hypotheses upon (a; b) in (1.2); (1.3); (1.4) are nothing but the same as those appearing in the definition the corresponding polynomials. Note that in (1.4) restriction in terms of conjugation as in (1.2) or (1.3) disappeared, a and b arising as limits of two pairs of conjugate coefficients, see relation (1.12). The p.d.f. corresponding to the weight-function f will be denoted by f˜. Note that whereas the weight function corresponding to a given family is subject to variations in the literature, up to a constant factor, the corresponding p.d.f. is unique. Probabilistic Askey scheme 5 Our Askey scheme for p.d.f.’s will therefore be the following. f˜a;b;c;d(x) (1.9) t y f˜a;b;c(x) f˜a;b(x) y % f˜(λ)(x; φ) f˜(α,β)(x) % y f˜(α)(x) Ó f˜0(x) In Proposition 2.1 relation (2.2) gives normalization constants as a function of one of the coefficient related to the corresponding orthogonal family. It enables us to give the explicit closed form of the p.d.f.’s for the seven families, see formulas (3.3),(4.4), (5.3),(6.2),(7.3), (8.3) and (9.3). Furthermore formulas (3.4); (4.5); (5.4); (6.2); (7.4); (8.4); (9.5) describe when necessary the asymptotic behavior of distribution tails providing as a corollary the domain of the moment generating function (m.g.f.). Concerning the latter, we will not give more detail as a closed form when it is available and the limit relations. We postpone this issue to a forthcoming paper. With obvious notations random variables with these distributions will be de- noted by Wil2(a; b; c; d); CDHahn2(a; b; c); CHahn(a; b; ); MeixPol(λ, φ); Jac(α; β); Lag(α); Her: Remark 1.2. Wilson and continuous dual Hahn polynomials are defined as poly- nomials of the variable x2. This is the reason why we consider the corresponding p.d.f as that of squared random variables. Remark 1.3. As mentioned above the random variables related to Jacobi, Laguerre and Hermite polynomials are closely related to the well-known Beta, Gamma and 6 J.-R. Pycke Gaussian densities respectively. More precisely, if these random variables and their densities are G(α + β) Beta(α; β) : xα(1 − x)β (0 < x < 1) G(α)G(β) βα Gamma(α; β) : xα−1e−βx (x > 0) G(α) 2 2 −1=2 − x N(µ, σ ) : (2π) e 2σ2 (x 2 R) it is readily checked that 1 − Jac(α − 1; β − 1) Lag(α − 1) Beta(α; β) = ; Gamma(α; β) = ; 2 β p N(µ, σ2) = µ + σ 2 Her: The Askey scheme for these random variables is the following. Wil2(a; b; c; d) (1.10) s w CDHahn2(a; b; c) CHahn(a; b) v ' MeixPol(λ, φ) Jac(α; β) ( v Lag(α) ! ~ Her It is shown in Proposition 2.2 that the sequence of moments of these random vari- ables is related to the sequence of polynomials via the elementary but far-reaching formula (2.3). In view of applications we give the first two or three polynomials of each family. They allow to compute the expectation and variance of our random variables, see formulas (3.2); (4.3); (5.2); (6.1); (7.2); (8.2); (9.4). The limit relations between p.d.f.’s corresponding to limit relations between polynomials are given by (3.5), (3.7), (3.8), (4.6), (5.5). (5.6), (6.3), (6.4), (7.5), (7.6) and (8.5). These convergences of densities, when combined with Lemma Probabilistic Askey scheme 7 2.3 which is a particular case of Scheffé’s Lemma, will imply in turn the following limit relations between our random variables.
Load more

Recommended publications
  • An Algebraic Interpretation of the Multivariate Q-Krawtchouk Polynomials
    An algebraic interpretation of the multivariate q-Krawtchouk polynomials The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Genest, Vincent X., Sarah Post, and Luc Vinet. “An Algebraic Interpretation of the Multivariate Q-Krawtchouk Polynomials.” The Ramanujan Journal (2016): n. pag. As Published http://dx.doi.org/10.1007/s11139-016-9776-2 Publisher Springer US Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/104790 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Ramanujan J DOI 10.1007/s11139-016-9776-2 An algebraic interpretation of the multivariate q-Krawtchouk polynomials Vincent X. Genest1 · Sarah Post2 · Luc Vinet3 Received: 31 August 2015 / Accepted: 18 January 2016 © Springer Science+Business Media New York 2016 Abstract The multivariate quantum q-Krawtchouk polynomials are shown to arise as matrix elements of “q-rotations” acting on the state vectors of many q-oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations, and recurrence relations. The extension to an arbitrary number of variables is presented. Keywords Multivariate q-Krawtchouk polynomials · q-Oscillator algebra · q-Rotations Mathematics Subject Classification 33D45 · 16T05 1 Introduction The purpose of this paper is to provide an algebraic model for the multi-variable q- Krawtchouk polynomials and to show how this model provides a cogent framework VXG is supported by a postdoctoral fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC).
    [Show full text]
  • Journal of Computational and Applied Mathematics Extensions of Discrete
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 225 (2009) 440–451 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Extensions of discrete classical orthogonal polynomials beyond the orthogonality R.S. Costas-Santos a,∗, J.F. Sánchez-Lara b a Department of Mathematics, University of California, South Hall, Room 6607 Santa Barbara, CA 93106, USA b Universidad Politécnica de Madrid, Escuela Técnica Superior de Arquitectura, Departamento de Matemática Aplicada, Avda Juan de Herrera, 4. 28040 Madrid, Spain article info a b s t r a c t α,β Article history: It is well-known that the family of Hahn polynomials fhn .xI N/gn≥0 is orthogonal with Received 5 November 2007 respect to a certain weight function up to degree N. In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can MSC: be characterized by a ∆-Sobolev orthogonality for every n and present a factorization for 33C45 Hahn polynomials for a degree higher than N. 42C05 We also present analogous results for dual Hahn, Krawtchouk, and Racah polynomials 34B24 and give the limit relations among them for all n 2 N0. Furthermore, in order to get Keywords: these results for the Krawtchouk polynomials we will obtain a more general property of Classical orthogonal polynomials orthogonality for Meixner polynomials. Inner product involving difference ' 2008 Elsevier B.V. All rights reserved.
    [Show full text]
  • Extensions of Discrete Classical Orthogonal Polynomials Beyond the Orthogonality
    W&M ScholarWorks Arts & Sciences Articles Arts and Sciences 2009 Extensions of discrete classical orthogonal polynomials beyond the orthogonality R. S. Costas-Santos J. F. Sanchez-Lara Follow this and additional works at: https://scholarworks.wm.edu/aspubs Recommended Citation Costas-Santos, R. S., & Sánchez-Lara, J. F. (2009). Extensions of discrete classical orthogonal polynomials beyond the orthogonality. Journal of Computational and Applied Mathematics, 225(2), 440-451. This Article is brought to you for free and open access by the Arts and Sciences at W&M ScholarWorks. It has been accepted for inclusion in Arts & Sciences Articles by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. Journal of Computational and Applied Mathematics 225 (2009) 440–451 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Extensions of discrete classical orthogonal polynomials beyond the orthogonality R.S. Costas-Santos a,∗, J.F. Sánchez-Lara b a Department of Mathematics, University of California, South Hall, Room 6607 Santa Barbara, CA 93106, USA b Universidad Politécnica de Madrid, Escuela Técnica Superior de Arquitectura, Departamento de Matemática Aplicada, Avda Juan de Herrera, 4. 28040 Madrid, Spain article info a b s t r a c t α,β Article history: It is well-known that the family of Hahn polynomials fhn .xI N/gn≥0 is orthogonal with Received 5 November 2007 respect to a certain weight function up to degree N. In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can MSC: be characterized by a ∆-Sobolev orthogonality for every n and present a factorization for 33C45 Hahn polynomials for a degree higher than N.
    [Show full text]
  • Image Analysis by Discrete Orthogonal Dual Hahn Moments
    Pattern Recognition Letters 28 (2007) 1688–1704 www.elsevier.com/locate/patrec Image analysis by discrete orthogonal dual Hahn moments Hongqing Zhu a, Huazhong Shu a,c,*, Jian Zhou a, Limin Luo a,c, J.L. Coatrieux b,c a Laboratory of Image Science and Technology, Department of Computer Science and Engineering, Southeast University, 210096 Nanjing, People’s Republic of China b INSERM U642, Laboratoire Traitement du Signal et de l’Image, Universite´ de Rennes I, 35042 Rennes, France c Centre de Recherche en Information Biome´dicale Sino-franc¸ais (CRIBs), France Received 29 May 2005; received in revised form 26 February 2007 Available online 29 April 2007 Communicated by L. Younes Abstract In this paper, we introduce a set of discrete orthogonal functions known as dual Hahn polynomials. The Tchebichef and Krawtchouk polynomials are special cases of dual Hahn polynomials. The dual Hahn polynomials are scaled to ensure the numerical stability, thus creating a set of weighted orthonormal dual Hahn polynomials. They are allowed to define a new type of discrete orthogonal moments. The discrete orthogonality of the proposed dual Hahn moments not only ensures the minimal information redundancy, but also elim- inates the need for numerical approximations. The paper also discusses the computational aspects of dual Hahn moments, including the recurrence relation and symmetry properties. Experimental results show that the dual Hahn moments perform better than the Legendre moments, Tchebichef moments, and Krawtchouk moments in terms of image reconstruction capability in both noise-free and noisy con- ditions. The dual Hahn moment invariants are derived using a linear combination of geometric moments.
    [Show full text]
  • The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its Q-Analogue
    The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Roelof Koekoek Ren´eF. Swarttouw Abstract We list the so-called Askey-scheme of hypergeometric orthogonal polynomials and we give a q- analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence rela- tion, the second order differential or difference equation, the forward and backward shift operator, the Rodrigues-type formula and generating functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give the limit relations between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation, second order difference equation, forward and backward shift operator, Rodrigues-type formula and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal poly- nomials. Finally, in chapter 5 we point out how the ‘classical’ hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues. Acknowledgement We would like to thank Professor Tom H. Koornwinder who suggested us to write a report like this. He also helped us solving many problems we encountered during the research and provided us with several references. Contents Preface 5 Definitions and miscellaneous formulas 7 0.1 Introduction . 7 0.2 The q-shifted factorials . 8 0.3 The q-gamma function and the q-binomial coefficient . 10 0.4 Hypergeometric and basic hypergeometric functions .
    [Show full text]
  • Doubling (Dual) Hahn Polynomials: Classification and Applications
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 003, 27 pages Doubling (Dual) Hahn Polynomials: Classification and Applications Roy OSTE and Joris VAN DER JEUGT Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium E-mail: [email protected], [email protected] Received July 13, 2015, in final form January 04, 2016; Published online January 07, 2016 http://dx.doi.org/10.3842/SIGMA.2016.003 Abstract. We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus trans- formations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester{Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models. Key words: Hahn polynomial; Racah polynomial; Christoffel pair; symmetric orthogonal polynomials; tridiagonal matrix; matrix eigenvalues; finite oscillator model 2010 Mathematics Subject Classification: 33C45; 33C80; 81R05; 81Q65 1 Introduction The tridiagonal (N + 1) × (N + 1) matrix of the following form 0 0 1 1 BN 0 2 C B C B N − 1 0 3 C C = B C (1.1) N+1 B .
    [Show full text]
  • Curriculum Vita
    CURRICULUM VITA NAME: Mourad E.H. Ismail AFFILIATION: University of Central Florida and King Saud University and ADDRESS: Department of Mathematics University of Central Florida 4000 Central Florida Blvd. P.O. Box 161364 Orlando, FL 32816-1364, USA. Telephones: Cell (407) 770-9959. POSITION: Research Professor at U of Central Florida PERSONAL DATA: Born April 27, 1944, in Cairo, Egypt. Male. Married. Canadian and Egyptian citizen, Permanent resident in the United States. EDUCATION: Ph.D. 1974 (Alberta), M.Sc. 1969 (Alberta), B.Sc. 1964 (Cairo). MATHEMATICAL GENEALOGY: Source: The Mathematics Genealogy Project, http://www.genealogy.ams.org • Yours truly, Ph. D. 1974, Advisor: Waleed Al-Salam. • Waleed Al-Salam, Ph. D. 1958, Advisor: Leonard Carlitz. • Leonard Carlitz, Ph. D. 1930, Advisor: Howard Mitchell. • Howard Mitchell, Ph. D. 1910, Advisor: Oswald Veblen. • Oswald Veblen, Ph. D. 1903, Advisor: Eliakim Hastings Moore. • Eliakim Hastings Moore, Ph. D. 1885, Advisor: H. A. Newton. • H. A. Newton, B. S. Yale 1850, Advisor: Michel Chasles. • Michel Chasles, Ph. D. 1814, Advisor: Simeon Poisson. • Simeon Poisson, Advisor: Joseph Lagrange. • Joseph Lagrange, Ph. D. Advisor: Leonhard Euler. 1 RESEARCH INTERESTS: Approximation theory, asymptotics, combinatorics, integral transforms and operational calculus, math- ematical physics, orthogonal polynomials and special functions. POSITIONS HELD: 2010{2012 Chair Professor, City University of Hong Kong 2003-2012 and 2013-present Professor, University of Central Florida 2008{2016 Distinguished Research
    [Show full text]
  • Spectral Transformations and Generalized Pollaczek Polynomials*
    METHODS AND APPLICATIONS OF ANALYSIS. © 1999 International Press Vol. 6, No. 3, pp. 261-280, September 1999 001 SPECTRAL TRANSFORMATIONS AND GENERALIZED POLLACZEK POLYNOMIALS* OKSANA YERMOLAYEVAt AND ALEXEI ZHEDANOV* To Richard Askey on his 65th birthday Abstract. The Christoffel and the Geronimus transformations of the classical orthogonal poly- nomials of a discrete variable are exploited to construct new families of the generalized Pollaczek polynomials. The recurrence coefficients wn, bn of these polynomials are rational functions of the argument n. The (positive) weight function is known explicitly. These polynomials are shown to belong to a subclass of the semi-classical orthogonal polynomials of a discrete variable. We illustrate the method by constructing a family of the modified Charlier polynomials which are orthogonal with respect to a perturbed Poisson distribution. The generating function of these polynomials provides a nontrivial extension of the class of the Meixner-Sheffer generating functions. 1. Introduction. The formal orthogonal polynomials Pn(x) are defined through the three-term recurrence relation ([8]) (1.1) iVfiOc) + unPn-i(x) + bnPn(x) = xPn(x) with the initial conditions (1.2) Po(x) = l, Pi(x)=x-bo. n n_1 The polynomials Pn(x) are monic (i.e. Pn{x) = x 4- 0(a; )). It can be shown that for arbitrary complex coefficients bn and (non-zero) un there exists a linear functional C such that k (1.3) £{Pn(x)x } = hn5nk, k<n, where (1.4) hn - u1U2"'Un ^ 0 are normalization constants. The functional £ is defined on the space of polynomials by its moments n (1.5) cn = £{x }, n = 0,l,---.
    [Show full text]
  • Charting the $ Q $-Askey Scheme
    Charting the q-Askey scheme Tom H. Koornwinder, [email protected] Dedicated to Jasper Stokman on the occasion of his fiftieth birthday, in admiration and friendship Abstract Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the q-Askey scheme together with their q-hypergeometric representations by k three sequences xk,hk,gk of Laurent polynomials in q , two of degree 1 and one of degree 2, satisfying certain constraints. This gives rise to a precise classification and parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also describe the four-manifold structure underlying the scheme. 1 Introduction The Askey scheme [1, p.46], [7, p.184] and the q-Askey scheme [7, p.414] display in a graphical way the families of (q-)hypergeomtric orthogonal polynomials as they occur as limit cases of the four-parameter top level families: Wilson and Racah polynomials for the Askey scheme and Askey–Wilson and q-Racah polynomials for the q-Askey scheme. By each arrow to the next lower level one parameter is lost. The bottom level families no longer depend on parameters. Since their introduction these schemes have been of great assistance to everybody who needs to do work with one or more of the families in the scheme. These schemes are also expected and partially proven to exist in other contexts, parallel to the original schemes or generalizing them. These contexts are: (i) (q-)hypergeometric biorthog- onal rational functions [3]; (ii) the nonsymmetric case [11], [12]; (iii) the q = −1 case starting arXiv:2108.03858v1 [math.CA] 9 Aug 2021 with the Bannai–Ito polynomials [2, pp.
    [Show full text]
  • Leonard Systems and Their Friends Jonathan Spiewak
    University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School March 2016 Leonard Systems and their Friends Jonathan Spiewak Follow this and additional works at: http://scholarcommons.usf.edu/etd Part of the Mathematics Commons Scholar Commons Citation Spiewak, Jonathan, "Leonard Systems and their Friends" (2016). Graduate Theses and Dissertations. http://scholarcommons.usf.edu/etd/6145 This Thesis is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Leonard Systems and their Friends by Jonathan Spiewak A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics College of Arts and Sciences University of South Florida Major Professor: Brian Curtin, Ph.D. Brendan Nagle, Ph.D. Jean-Fran¸coisBiasse, Ph.D. Date of Approval: March 2, 2016 Keywords: Leonard pair, Split decomposition, equitable sl 2 basis, equitable Uq(sl 2) generators Copyright c 2016, Jonathan Spiewak Acknowledgements I would like to sincerely thank Dr. Brian Curtin for his knowledge, guidance, and patience. I would also like to thank the Department of Mathematics at University of South Florida, my thesis committee, and Dr. Ana Torres for her interest and support. I would like to thank my parents, for believing in me and always encouraging me to aspire high. I would like to thank my husband, Kenny, who kept me focused and motivated throughout this entire process. Table of Contents List of Tables .
    [Show full text]
  • Orthogonal Polynomials and Special Functions
    Orthogonal Polynomials and Special Functions SIAM Activity Group on Orthogonal Polynomials and Special Functions ???? Newsletter ???? Published Three Times a Year October 1997 Volume 8, Number 1 Contents formatik, Mathematik und Naturwissenschaften). My new address there is: From the Editor . 1 Prof. Dr. Wolfram Koepf Message from the Chair: Charter Renewal . 2 Fachbereich IMN Reports from Meetings and Conferences . 4 HTWK Leipzig Forthcoming Meetings and Conferences . 5 Gustav-Freytag-Str. 42 A Books and Journals . 9 D-04277 Leipzig Call For Papers . 16 phone: +49-341-307 64 95 Software Announcements . 17 fax: +49-341-301 27 22 Problems and Solutions . 17 e-mail: [email protected] Miscellaneous . 19 WWW: http://www.imn.htwk-leipzig.de/~koepf How to Contribute to the Newsletter . 24 Activity Group: Addresses . 24 At least for the next few months, I will, however, be still a±liated with Konrad-Zuse-Zentrum so that you can send your messages to both places, L P O N A L Y and my old WWW homepage will still be valid. O N G O O M These changes are time-consuming, and I am H I A T L R SIAM indebted to Tom Koornwinder and Martin Mul- S O Activity Group doon who do a great job as Editors of the elec- tronic newsletter OP-SF Net. Fortunately I could S S P E Est. 1990 N borrow much material from them. Indeed, their C O I I A C T material was the main source for my collection L F U N today. I would like to announce that I will resign my From the Editor editorship at the end of the period of my appoint- ment, i.e.
    [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]