DOCSLIB.ORG

Orthogonal Polynomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Orthogonal Polynomials Orthogonal polynomials We start with 1 De¯nition 1. A sequence of polynomials fpn(x)gn=0 with degree[pn(x)] = n for each n is called orthogonal with respect to the weight function w(x) on the interval (a; b) with a < b if ( Z b 0; m 6= n w(x)pm(x)pn(x) dx = hn ±mn with ±mn := a 1; m = n: The weight function w(x) should be continuous and positive on (a; b) such that the moments Z b n ¹n := w(x) x dx; n = 0; 1; 2;::: a exist. Then the integral Z b hf; gi := w(x)f(x)g(x) dx a denotes an inner product of the polynomials f and g. The interval (a; b) is called the interval of orthogonality. This interval needs not to be ¯nite. If hn = 1 for each n 2 f0; 1; 2;:::g the sequence of polynomials is called orthonormal, and if n pn(x) = knx + lower order terms with kn = 1 for each n 2 f0; 1; 2;:::g the polynomials are called monic. Example. As an example we take w(x) = 1 and (a; b) = (0; 1). Using the Gram-Schmidt process the orthogonal polynomials can be constructed as follows. Start with the sequence 2 f1; x; x ;:::g. Choose p0(x) = 1. Then we have hx; p0(x)i hx; 1i 1 p1(x) = x ¡ p0(x) = x ¡ = x ¡ ; hp0(x); p0(x)i h1; 1i 2 since Z Z 1 1 1 h1; 1i = dx = 1 and hx; 1i = x dx = : 0 0 2 Further we have 2 2 2 hx ; p0(x)i hx ; p1(x)i p2(x) = x ¡ p0(x) ¡ p1(x) hp0(x); p0(x)i hp1(x); p1(x)i 2 2 1 µ ¶ µ ¶ 2 hx ; 1i hx ; x ¡ 2 i 1 2 1 1 2 1 = x ¡ ¡ 1 1 x ¡ = x ¡ ¡ x ¡ = x ¡ x + ; h1; 1i hx ¡ 2 ; x ¡ 2 i 2 3 2 6 since Z 1 Z 1 µ ¶ 2 2 1 2 1 2 1 1 1 1 hx ; 1i = x dx = ; hx ; x ¡ 2 i = x x ¡ dx = ¡ = 0 3 0 2 4 6 12 and Z 1 µ ¶2 Z 1 µ ¶ 1 1 1 2 1 1 1 1 1 hx ¡ 2 ; x ¡ 2 i = x ¡ dx = x ¡ x + dx = ¡ + = : 0 2 0 4 3 2 4 12 1 2 1 The polynomials p0(x) = 1, p1(x) = x ¡ 2 and p2(x) = x ¡ x + 6 are the ¯rst three monic orthogonal polynomials on the interval (0; 1) with respect to the weight function w(x) = 1. 1 Repeating this process we obtain 3 3 1 9 2 1 p (x) = x3 ¡ x2 + x ¡ ; p (x) = x4 ¡ 2x3 + x2 ¡ x + ; 3 2 5 20 4 7 7 70 5 20 5 5 1 p (x) = x5 ¡ x4 + x3 ¡ x2 + x ¡ ; 5 2 9 6 42 252 and so on. p The orthonormal polynomials would be q0(x) = p0(x)= h0 = 1, µ ¶ p p p p2(x) 2 1 q1(x) = p1(x)= h1 = 2 3(x ¡ 1=2); q2(x) = p = 6 5 x ¡ x + ; h2 6 µ ¶ p p3(x) 3 3 2 3 1 q3(x) = p = 20 7 x ¡ x + x ¡ ; h3 2 5 20 etcetera. All sequences of orthogonal polynomials satisfy a three term recurrence relation: 1 Theorem 1. A sequence of orthogonal polynomials fpn(x)gn=0 satis¯es pn+1(x) = (Anx + Bn)pn(x) + Cnpn¡1(x); n = 1; 2; 3;:::; where kn+1 An hn An = ; n = 0; 1; 2;::: and Cn = ¡ ¢ ; n = 1; 2; 3;:::: kn An¡1 hn¡1 Proof. Since degree [pn(x)] = n for each n 2 f0; 1; 2;:::g the sequence of orthogonal polyno- 1 mials fpn(x)gn=0 is linearly independent. Let An = kn+1=kn. Then pn+1(x) ¡ Anxpn(x) is a polynomial of degree · n. Hence Xn pn+1(x) ¡ Anxpn(x) = ckpk(x): k=0 The orthogonality property now gives Xn hpn+1(x) ¡ Anxpn(x); pk(x)i = cmhpm(x); pk(x)i = ckhpk(x); pk(x)i = ck hk: m=0 This implies hk ck = hpn+1(x) ¡ Anxpn(x); pk(x)i = hpn+1(x); pk(x)i ¡ Anhxpn(x); pk(x)i = ¡Anhpn(x); xpk(x)i: For k < n ¡ 1 we have degree [xpk(x)] < n which implies that hpn(x); xpk(x)i = 0. Hence: ck = 0 for k < n ¡ 1. This proves that the polynomials satisfy the three term recurrence relation pn+1(x) ¡ Anxpn(x) = cnpn(x) + cn¡1pn¡1(x); n = 1; 2; 3;:::: Further we have kn¡1 An hn hn¡1 cn¡1 = ¡Anhpn(x); xpn¡1(x)i = ¡An hn =) cn¡1 = ¡ ¢ : kn An¡1 hn¡1 This proves the theorem. 2 Note that the three term recurrence relation for a sequence of monic (kn = 1) orthogonal 1 polynomials fpn(x)gn=0 has the form hn pn+1(x) = xpn(x) + Bnpn(x) + Cnpn¡1(x) with Cn = ¡ ; n = 1; 2; 3;:::: hn¡1 A consequence of the three term recurrence relation is 1 Theorem 2. A sequence of orthogonal polynomials fpn(x)gn=0 satis¯es Xn p (x)p (y) k p (x)p (y) ¡ p (y)p (x) k k = n ¢ n+1 n n+1 n ; n = 0; 1; 2;::: (1) hk hn kn+1 x ¡ y k=0 and Xn 2 fpk(x)g kn ¡ 0 0 ¢ = ¢ pn+1(x)pn(x) ¡ pn+1(x)pn(x) ; n = 0; 1; 2;:::: (2) hk hn kn+1 k=0 Formula (1) is called the Christo®el-Darboux formula and (2) its confluent form. Proof. The three term recurrence relation implies that pn+1(x)pn(y) = (Anx + Bn)pn(x)pn(y) + Cnpn¡1(x)pn(y) and pn+1(y)pn(x) = (Any + Bn)pn(y)pn(x) + Cnpn¡1(y)pn(x): Subtraction gives pn+1(x)pn(y) ¡ pn+1(y)pn(x) = An(x ¡ y)pn(x)pn(y) + Cn [pn¡1(x)pn(y) ¡ pn¡1(y)pn(x)] : This leads to the telescoping sum Xn p (x)p (y) Xn p (x)p (y) ¡ p (y)p (x) (x ¡ y) k k = k+1 k k+1 k hk Ak hk k=1 k=1 Xn p (x)p (y) ¡ p (y)p (x) ¡ k k¡1 k k¡1 Ak¡1 hk¡1 k=1 p (x)p (y) ¡ p (y)p (x) k2(x ¡ y) = n+1 n n+1 n ¡ 0 : An hn h0 This implies that Xn p (x)p (y) p (x)p (y) ¡ p (y)p (x) (x ¡ y) k k = n+1 n n+1 n hk An hn k=0 kn = (pn+1(x)pn(y) ¡ pn+1(y)pn(x)) ; hn kn+1 which proves (1). The confluent form (2) then follows by taking the limit y ! x: p (x)p (y) ¡ p (y)p (x) lim n+1 n n+1 n y!x x ¡ y p (x)(p (x) ¡ p (y)) ¡ p (x)(p (x) ¡ p (y)) = lim n n+1 n+1 n+1 n n y!x x ¡ y 0 0 = pn(x)pn+1(x) ¡ pn+1(x)pn(x): 3 Zeros 1 Theorem 3. If fpn(x)gn=0 is a sequence of orthogonal polynomials on the interval (a; b) with respect to the weight function w(x), then the polynomial pn(x) has exactly n real simple zeros in the interval (a; b). Proof. Since degree[pn(x)] = n the polynomial has at most n real zeros. Suppose that pn(x) has m · n distinct real zeros x1; x2; : : : ; xm in (a; b) of odd order (or multiplicity). Then the polynomial pn(x)(x ¡ x1)(x ¡ x2) ¢ ¢ ¢ (x ¡ xm) does not change sign on the interval (a; b). This implies that Z b w(x)pn(x)(x ¡ x1)(x ¡ x2) ¢ ¢ ¢ (x ¡ xm) dx 6= 0: a By orthogonality this integral equals zero if m < n. Hence: m = n, which implies that pn(x) has n distinct real zeros of odd order in (a; b). This proves that all n zeros are distinct and simple (have order or multiplicity equal to one). 1 Theorem 4. If fpn(x)gn=0 is a sequence of orthogonal polynomials on the interval (a; b) with respect to the weight function w(x), then the zeros of pn(x) and pn+1(x) separate each other. Proof. This follows from the confluent form (2) of the Christo®el-Darboux formula. Note that Z b 2 hn = w(x) fpn(x)g dx > 0; n = 0; 1; 2;:::: a This implies that Xn 2 kn ¡ 0 0 ¢ fpk(x)g ¢ pn+1(x)pn(x) ¡ pn+1(x)pn(x) = > 0: hn kn+1 hk k=0 Hence kn ¡ 0 0 ¢ ¢ pn+1(x)pn(x) ¡ pn+1(x)pn(x) > 0: kn+1 Now suppose that xn;k and xn;k+1 are two consecutive zeros of pn(x) with xn;k < xn;k+1. 0 0 Since all n zeros of pn(x) are real and simple pn(xn;k) and pn(xn;k+1) should have opposite signs. Hence we have 0 0 pn(xn;k) = 0 = pn(xn;k+1) and pn(xn;k) ¢ pn(xn;k+1) < 0: This implies that pn+1(xn;k) ¢ pn+1(xn;k+1) should be negative too. Then the continuity of pn+1(x) implies that there should be at least one zero of pn+1(x) between xn;k and xn;k+1.
Load more

Recommended publications
  • Operational Techniques for Laguerre and Legendre Polynomials
    INTERNATIONAL JOURNAL of PURE MATHEMATICS Volume 2, 2015 Operational techniques for Laguerre and Legendre polynomials Clemente Cesarano, Dario Assante, A. R. El Dhaba Abstract— By starting from the concepts and the related n , (2) formalism of the Monomiality Principle, we exploit methods of δ xn =∏() xm − δ operational nature to describe different families of Laguerre m=0 polynomials, ordinary and generalized, and to introduce the Legendre polynomials through a special class of Laguerre polynomials whose associated multiplication and derivative operators read: themselves. Many of the identities presented, involving families of d different polynomials were derived by using the structure of the ^ δ operators who satisfy the rules of a Weyl group. M= xe dx In this paper, we first present the Laguerre and Legendre d δ −1 (3) polynomials, and their generalizations, from an operational point of ^ e dx P = . view, we discuss some operational identities and further we derive δ some interesting relations involving an exotic class of orthogonal polynomials in the description of Legendre polynomials. It is worth noting that, when δ = 0 then: Keywords— Laguerre polynomials, Legendre polynomials, Weyl group, Monomiality Principle, Generating Functions. ^ Mx= (4) ^ d I. INTRODUCTION P = . In this section we will present the concepts and the related aspects of dx the Monomiality Principle [1,2] to explore different approaches for More in general, a given polynomial Laguerre and Legendre polynomials [3]. The associated operational calculus introduced by the Monomiality Principle allows us to reformulate the theory of the generalized Laguerre and Legendre pxn (), nx∈∈, polynomials from a unified point of view. In fact, these are indeed shown to be particular cases of more general polynomials and can be can be considered a quasi-monomial if two operators also used to derive classes of isospectral problems, an interesting example is given by the generalized Bessel functions [4].
    [Show full text]
  • Hermite Functions and the Quantum Oscillator REVISE
    Hermite Functions and the Quantum Oscillator Background Differentials Equations handout The Laplace equation – solution and application Concepts of primary interest: Separation constants Sets of orthogonal functions Discrete and continuous eigenvalue spectra Generating Functions Rodrigues formula representation Recurrence relations Raising and lowering operators Sturm-Liouville Problems Sample calculations: Tools of the trade: k ½ REVISE: Use the energy unit ( /m) and include the roots of 2 from the beginning. The Quantum Simple Harmonic Oscillator is one of the problems that motivate the study of the Hermite polynomials, the Hn(x). Q.M.S. (Quantum Mechanics says.): 2 du2 n ()()01 kx2 E u x [Hn.1] 2mdx2 2 nn This equation is to be attacked and solved by the numbers. STEP ONE: Convert the problem from one in physics to one in mathematics. The equation as written has units of energy. The constant has units of energy * time, m 2 has units of mass, and k has units of energy per area. Combining, has units of mk Contact: [email protected] 1/4 4 mk -1 (length) so a dimensioned scaling constant = 2 is defined with units (length) as a step toward defining a dimensionless variable z = x. The equation itself is k ½ divided by the natural unit of energy ½ ( /m) leading to a dimensionless constant 2 En n = which is (proportional to) the eigenvalue, the energy in natural units. (/km )1/2 The equation itself is now expressed in a dimensionless (mathspeak) form: du2 n ()()0 zuz2 . [Hn.2] (**See problem 2 for important details.) dz2 nn The advantage of this form is that one does not need to write down as many symbols.
    [Show full text]
  • ' Unit 16 Her-Mite and Laguerre Polynomials
    ' UNIT 16 HER-MITE AND LAGUERRE POLYNOMIALS Structure Introduction Objectives Hermite's Differential Equation and ~eAitePolynomials Generating Function and Recurrence Relations for Hermite Polynomials Orthogonality Relations for Hermite Polynomials One Dimensional Harmonic Oscillator Laguerre's Differentiation Equation and Laguerre Polynomials Generating Polynomials and Recurrence Relations for Laguerre Polynomials Orthogonality Relations for Laguerre Polynomials Summary Terminal Questions Solutions and Answers 5 16.1 INTRODUCTION You have studied the properties of the Legendre polynomials, and the Bessel functions in Units 14 and 15, respectively. In this unit, you will learn the properties of Hermite polynomials and Laguerre polynomials. These polynomials are useful in studying the behaviour of a harmonic oscillator and the hydrogen atom using quantum mechanics. In fact, these polynomials enable us to describe the microscopic world of molecules, atoms, and sub- atomic particles in a mathematical language. A study of the properties of these polynomials and a knowledge of their zeros is also helpful in developing numerical methods. Like Legendre and Bessel polynomials, the Hermite and Laguerre polynomials are solutions of linear second order ordinary differential equations (ODE) known as the Hermite and the Laguerre differential equations. You will come across these ODES and the two polynomials in Sec. 16.2 and 16.6, respectively. The generating function and the recurrence relations for Hermite polynomials are given in Sec. 16.3. You have already seen the importance of the generating functions for Legendre and Bessel polynomials in the last units. You will now learn how the generating function for Hermite polynomials is helpful in obtaining many of their properties. The recurrence relations connecting Hermite polynomials of different orders and their derivatives are derived from the generating function in the same section.
    [Show full text]
  • HERMITE POLYNOMIALS Schrödinger's Equation for A
    MATH 105B EXTRA/REVIEW TOPIC: HERMITE POLYNOMIALS PROF. MICHAEL VANVALKENBURGH Schr¨odinger'sequation for a quantum harmonic oscillator is 1 d2 − + x2 u = u. 2 dx2 This says u is an eigenfunction with eigenvalue . In physics, represents an energy value. Let 1 d a = p x + the \annihilation" operator, 2 dx and 1 d ay = p x − the \creation" operator. 2 dx Note that, as an operator, d2 aya + aay = − + x2; dx2 which we can use to rewrite Schr¨odinger'sequation. Fact. The function −1=4 −x2=2 u0(x) = π e represents a quantum particle in its \ground state." (Its square is the probability density function of a particle in its lowest energy state.) Note that au0(x) = 0; which justifies calling the operator a the \annihilation" operator! On the other hand, y u1(x) : = a u0(x) 2 = π−1=42−1=2(2x)e−x =2 represents a quantum particle in its next highest energy state. So the operator ay \creates" one unit of energy! 1 MATH 105B EXTRA/REVIEW TOPIC: HERMITE POLYNOMIALS 2 Each application of ay creates a unit of energy, and we get the \Hermite functions" −1=2 y n un(x) : = (n!) (a ) u0(x) −1=4 −n=2 −1=2 −x2=2 = π 2 (n!) Hn(x)e ; where Hn(x) is a polynomial of order n called \the nth Hermite polynomial." In particular, H0(x) = 1 and H1(x) = 2x. Fact: Hn satisfies the ODE d2 d − 2x + 2n H = 0: dx2 dx n For review, let's use the power series method: We look for Hn of the form 1 X k Hn(x) = Akx : k=0 We plug in and compare coefficients to find: A2 = −nA0 2(k − n) A = A for k ≥ 1: k+2 (k + 2)(k + 1) k Note how there is a natural split into odd and even terms, similar to what happened for Legendre's equation in x5.2.
    [Show full text]
  • Chebyshev Polynomials of the Second, Third and Fourth Kinds in Approximation, Indefinite Integration, and Integral Transforms *
    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 49 (1993) 169-178 169 North-Holland CAM 1429 Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms * J.C. Mason Applied and Computational Mathematics Group, Royal Military College of Science, Shrivenham, Swindon, Wiltshire, United Kingdom Dedicated to Dr. D.F. Mayers on the occasion of his 60th birthday Received 18 February 1992 Revised 31 March 1992 Abstract Mason, J.C., Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms, Journal of Computational and Applied Mathematics 49 (1993) 169-178. Chebyshev polynomials of the third and fourth kinds, orthogonal with respect to (l+ x)‘/‘(l- x)-‘I* and (l- x)‘/*(l+ x)-‘/~, respectively, on [ - 1, 11, are less well known than traditional first- and second-kind polynomials. We therefore summarise basic properties of all four polynomials, and then show how some well-known properties of first-kind polynomials extend to cover second-, third- and fourth-kind polynomials. Specifically, we summarise a recent set of first-, second-, third- and fourth-kind results for near-minimax constrained approximation by series and interpolation criteria, then we give new uniform convergence results for the indefinite integration of functions weighted by (1 + x)-i/* or (1 - x)-l/* using third- or fourth-kind polynomial expansions, and finally we establish a set of logarithmically singular integral transforms for which weighted first-, second-, third- and fourth-kind polynomials are eigenfunctions.
    [Show full text]
  • Generalizations of Chebyshev Polynomials and Polynomial Mappings
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 10, October 2007, Pages 4787–4828 S 0002-9947(07)04022-6 Article electronically published on May 17, 2007 GENERALIZATIONS OF CHEBYSHEV POLYNOMIALS AND POLYNOMIAL MAPPINGS YANG CHEN, JAMES GRIFFIN, AND MOURAD E.H. ISMAIL Abstract. In this paper we show how polynomial mappings of degree K from a union of disjoint intervals onto [−1, 1] generate a countable number of spe- cial cases of generalizations of Chebyshev polynomials. We also derive a new expression for these generalized Chebyshev polynomials for any genus g, from which the coefficients of xn can be found explicitly in terms of the branch points and the recurrence coefficients. We find that this representation is use- ful for specializing to polynomial mapping cases for small K where we will have explicit expressions for the recurrence coefficients in terms of the branch points. We study in detail certain special cases of the polynomials for small degree mappings and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an explicit expression for the discrimi- nant for the genus 1 case of our Chebyshev polynomials that is valid for any configuration of the branch point. 1. Introduction and preliminaries Akhiezer [2], [1] and, Akhiezer and Tomˇcuk [3] introduced orthogonal polynomi- als on two intervals which generalize the Chebyshev polynomials. He observed that the study of properties of these polynomials requires the use of elliptic functions. In thecaseofmorethantwointervals,Tomˇcuk [17], investigated their Bernstein-Szeg˝o asymptotics, with the theory of Hyperelliptic integrals, and found expressions in terms of a certain Abelian integral of the third kind.
    [Show full text]
  • Chebyshev and Fourier Spectral Methods 2000
    Chebyshev and Fourier Spectral Methods Second Edition John P. Boyd University of Michigan Ann Arbor, Michigan 48109-2143 email: [email protected] http://www-personal.engin.umich.edu/jpboyd/ 2000 DOVER Publications, Inc. 31 East 2nd Street Mineola, New York 11501 1 Dedication To Marilyn, Ian, and Emma “A computation is a temptation that should be resisted as long as possible.” — J. P. Boyd, paraphrasing T. S. Eliot i Contents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions ................................ 10 1.8 Non-Interpolating and Pseudospectral ...................... 12 1.9 Nonlinearity ..................................... 13 1.10 Time-dependent problems ............................. 15 1.11 FAQ: Frequently Asked Questions ........................ 16 1.12 The Chrysalis .................................... 17 2 Chebyshev & Fourier Series 19 2.1 Introduction ..................................... 19 2.2 Fourier series .................................... 20 2.3 Orders of Convergence ............................... 25 2.4 Convergence Order ................................. 27 2.5 Assumption of Equal Errors ...........................
    [Show full text]
  • Dymore User's Manual Chebyshev Polynomials
    Dymore User's Manual Chebyshev polynomials Olivier A. Bauchau August 27, 2019 Contents 1 Definition 1 1.1 Zeros and extrema....................................2 1.2 Orthogonality relationships................................3 1.3 Derivatives of Chebyshev polynomials..........................5 1.4 Integral of Chebyshev polynomials............................5 1.5 Products of Chebyshev polynomials...........................5 2 Chebyshev approximation of functions of a single variable6 2.1 Expansion of a function in Chebyshev polynomials...................6 2.2 Evaluation of Chebyshev expansions: Clenshaw's recurrence.............7 2.3 Derivatives and integrals of Chebyshev expansions...................7 2.4 Products of Chebyshev expansions...........................8 2.5 Examples.........................................9 2.6 Clenshaw-Curtis quadrature............................... 10 3 Chebyshev approximation of functions of two variables 12 3.1 Expansion of a function in Chebyshev polynomials................... 12 3.2 Evaluation of Chebyshev expansions: Clenshaw's recurrence............. 13 3.3 Derivatives of Chebyshev expansions.......................... 14 4 Chebychev polynomials 15 4.1 Examples......................................... 16 1 Definition Chebyshev polynomials [1,2] form a series of orthogonal polynomials, which play an important role in the theory of approximation. The lowest polynomials are 2 3 4 2 T0(x) = 1;T1(x) = x; T2(x) = 2x − 1;T3(x) = 4x − 3x; T4(x) = 8x − 8x + 1;::: (1) 1 and are depicted in fig.1. The polynomials can be generated from the following recurrence rela- tionship Tn+1 = 2xTn − Tn−1; n ≥ 1: (2) 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 CHEBYSHEV POLYNOMIALS −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 XX Figure 1: The seven lowest order Chebyshev polynomials It is possible to give an explicit expression of Chebyshev polynomials as Tn(x) = cos(n arccos x): (3) This equation can be verified by using elementary trigonometric identities.
    [Show full text]
  • New Formulae of Squares of Some Jacobi Polynomials Via Hypergeometric Functions
    Hacettepe Journal of Mathematics and Statistics Volume 46 (2) (2017), 165 176 New formulae of squares of some Jacobi polynomials via hypergeometric functions W.M. Abd- Elhameed∗ y Abstract In this article, a new formula expressing explicitly the squares of Jacobi polynomials of certain parameters in terms of Jacobi polynomials of ar- bitrary parameters is derived. The derived formula is given in terms of ceratin terminating hypergeometric function of the type 4F3(1). In some cases, this 4F3(1) can be reduced by using some well-known re- duction formulae in literature such as Watson's and Pfa-Saalschütz's identities. In some other cases, this 4F3(1) can be reduced by means of symbolic computation, and in particular Zeilberger's, Petkovsek's and van Hoeij's algorithms. Hence, some new squares formulae for Jacobi polynomials of special parameters can be deduced in reduced forms which are free of any hypergeometric functions. Keywords: Jacobi polynomials; linearization coecients; generalized hypergeo- metric functions; computer algebra, standard reduction formulae 2000 AMS Classication: 33F10; 33C20; 33Cxx; 68W30 Received : 23.02.2016 Accepted : 19.05.2016 Doi : 10.15672/HJMS.20164518618 1. Introduction The Jacobi polynomials are of fundamental importance in theoretical and applied mathematical analysis. The class of Jacobi polynomials contains six well-known fami- lies of orthogonal polynomials, they are, ultraspherical, Legendre and the four kinds of Chebyshev polynomials. The Jacobi polynomials in general and their six special polyno- mials in particular are extensively employed in obtaining numerical solutions of ordinary, fractional and partial dierential equations. In this respect, these polynomials are em- ployed for the sake of obtaining spectral solutions for various kinds of dierential equa- tions.
    [Show full text]
  • Some Classical Multiple Orthogonal Polynomials
    Some classical multiple orthogonal polynomials ∗ Walter Van Assche and Els Coussement Department of Mathematics, Katholieke Universiteit Leuven 1 Classical orthogonal polynomials One aspect in the theory of orthogonal polynomials is their study as special functions. Most important orthogonal polynomials can be written as terminating hypergeometric series and during the twentieth century people have been working on a classification of all such hypergeometric orthogonal polynomial and their characterizations. The very classical orthogonal polynomials are those named after Jacobi, Laguerre, and Hermite. In this paper we will always be considering monic polynomials, but in the literature one often uses a different normalization. Jacobi polynomials are (monic) polynomials of degree n which are orthogonal to all lower degree polynomials with respect to the weight function (1 x)α(1+x)β on [ 1, 1], where α,β > 1. The change of variables x 2x 1 gives Jacobi polynomials− on [0, 1]− for the weight function− w(x) = xβ(1 x)α, and we7→ will− denote (α,β) − these (monic) polynomials by Pn (x). They are defined by the orthogonality conditions 1 P (α,β)(x)xβ(1 x)αxk dx =0, k =0, 1,...,n 1. (1.1) n − − Z0 (α) The monic Laguerre polynomials Ln (x) (with α > 1) are orthogonal on [0, ) to all − α x ∞ polynomials of degree less than n with respect to the weight w(x)= x e− and hence satisfy the orthogonality conditions ∞ (α) α x k Ln (x)x e− x dx =0, k =0, 1,...,n 1. (1.2) 0 − arXiv:math/0103131v1 [math.CA] 21 Mar 2001 Z Finally, the (monic) Hermite polynomials Hn(x) are orthogonal to all lower degree polyno- x2 mials with respect to the weight function w(x)= e− on ( , ), so that −∞ ∞ ∞ x2 k H (x)e− x dx =0, k =0, 1,...,n 1.
    [Show full text]
  • 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.
    [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]