DOCSLIB.ORG

Associated Legendre Polynomial

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Associated Legendre Polynomial Mathematical Methods 2 (PART I) Lecturer: Dr. D.J. Miller Room 535, Kelvin Building [email protected] Location: Room 312, Kelvin Building Recommended Text: Boas, 3 rd Edition 1 Outline 1. Curvilinear coordinate systems 2. Partial Differential Equations in cylindrical and spherical polar coordinates 3. Legendre Polynomials and Bessel Functions 4. Hermite and Laguerre Polynomials 5. Gamma and Beta Functions, and Stirling’s approximation 2 1. Curvilinear coordinate systems 1.1 Revision of vector calculus with Cartesian coordinates Consider a 3-dimensional vector space. Any point in the space can be written in terms of three coordinates ___________ and three basis vectors Sometimes the coordinates are written as _______ and the unit vectors as (I will use both notations, but avoid i, j and k). Any vector A in this space, can be written in terms of these unit vectors as, I will sometimes assume the Einstein Summation convention and omit the summation sign Σ when I have two repeated indices (unless otherwise specified), e.g. 3 These unit vectors are orthonormal (orthogonal and of unit length) if Kronecker delta The position vector is given by remember Einstein r summation convention! The scalar product of two vectors A and B is The vector product of two vectors A and B is this has a sum over i, j and k where is the Levi Civita symbol , which is +1 for an even permutation of 1, 2 ǫijk and 3 and -1 for an odd permutation, and zero if any of the indices are the same, e.g. 4 For example, the z-component is: The gradient operator ∇ is given by From this, and the definitions above, follow the gradient of a scalar field φ(r), one number for each value of r and the divergence and curl of a vector field A(r), three numbers for each value of r 5 Exercise : Using the Einstein summation convention, prove (you will need for the this one), and 6 1.2 Curvilinear coordinates in physics problems Key Point: Understand the importance of curvilinear coordinates in solving physics problems Very often, physical systems are described by classical (or quantum) field theories. We have a field, e.g. , which obeys a differential equation over all space, e.g. the Schrödinger Equation The field is constrained by the differential equation and by boundary conditions . The differential equation is independent of coordinate system (though may look very different in different coordinate systems) The boundary condition is often very much simpler in one particular choice of coordinates, making the solution simpler in these coordinates. Use symmetries of the boundary conditions (incl. the position of sources/sinks) to choose your coordinate system. 7 Field Differential Equation General Solution scenario coordinate independent system Boundary Specific Conditions Solution scenario dependent 8 Example: Spherical Polar Coordinates Describe the position of a point in space using the distance from the origin, r, and two angles, θ and φ. This is useful in problems with spherical symmetry . e.g. Consider a hollow sphere or radius R kept at a constant temperature T. The temperature inside the sphere obeys Laplace’s equation, which is coordinate system independent. However, the boundary condition is much more easily expressed in spherical polar coordinates: 9 Example: Cylindrical Coordinates Describe the position of a point in space using the distance along the z-axis, the distance from the z- axis, r, and an angle, θ. This is useful in problems with cylindrical symmetry . e.g. Consider a wire with a uniform charge density ρ0. The electric potential generated by the charge distribution is given by Poisson’s equation, which is coordinate system independent (and becomes Laplace’s equation away from the charge). However, the charge density is much more easily expressed in cylindrical polar coordinates (with the wire running along the z-axis): 10 1.3 Unit vectors and scale factors Key Point: Write down the definition of basis vectors and scale factors for general curvilinear coordinates In Cartesian coordinates, if I change the x-coordinate of the position vector r by an amount dx , then the object has moved a distance dx . If I change all of the coordinates (infinitesimally) at once, I move r a distance dr where using Pythagoras’ theorem. dr is know as the infinitesimal line element . This may seem obvious, but the analogue is not true in general curvilinear coordinates . 11 e.g. Imagine changing θ by and amount d θ in cylindrical coordinates: The distance moved is for dθ small. (The z-direction is out of the slide.) How far we move depends on the value of the coordinate r. How far we move for a particular coordinate shift is known as a scale factor . Also notice that the direction we move depends on the coordinate θ. The direction we move for a particular coordinate shift is given by the basis vector . 12 Imagine a general coordinate system described by ( =1,2,3) and related to Cartesians qi i via As we change the coordinate , the position vector will move: qi r (no sum over i) basis vector scale factor The scale factor is the magnitude of the vector , i.e. The basis vector is the unit vector in the direction of , i.e. (no sum) 13 As before, the basis vectors are orthogonal if Let’s call the distance moved by changing the coordinate by an amount ds i qi dq i (no sum) For orthogonal coordinates, we can use Pythogoras’ theorem again to write the total displacement from altering all three coordinates: More formally, the change in the position vector is So, 14 More generally, one may define the metric of the space according to gij , metric (no sum) So, in our case we have , but this can in principle be more complicated. The volume element is given by ds ds ds 15 Key Point: Derive the scale factors and unit vectors for various coordinate systems (no sum) Example : Cartesian Coordinates (trivial!) The position vector is So, These are already unit vectors, so are the basis vectors and the scale factors are, The volume element is , and the square of the infinitesimal line element is . 16 Example : Spherical Polar Coordinates They are related to Cartesians by: and the position vector is ( but since changes with and , it is more convenient to use , and .) er θ φ ex ey ez So, 17 Normalize these vectors to find the basis vectors and scale factors: So, we have scale factors : , , . basis vectors : The volume element is , and the square of the infinitesimal line element is . 18 We could have derived the scale factors directly from the square of the infinitesimal line element: So, write ds =dx +dy +dz and plug in dx , dy and dz (above). The cross terms vanish and we are left with The lack of cross-terms tells us the coordinates are orthogonal, and the coefficients of the diagonal terms are the square of the scale factors. Exercise : Find the scale factors, unit vectors, volume element and infinitesimal line element-squared for cylindrical coordinates. 19 1.3 The Gradient Key Point: Derive an expression for the gradient in general orthogonal curvilinear co-ordinate systems The gradient in Cartesian coordinates was: This tells us how fast a field changes with position, i.e. with respect to the distance moved NOT how much the parameter changes ⇒ We must take into account the scale factors. For a curvilinear coordinates , recall the distance moved for a change was qi dq i ds = h dq , so the component of the gradient in the direction of e is i qi i qi (no sum) and the gradient is 20 1.5 The Divergence, Curl and Laplacian Operators Key Point: Derive expressions for div, grad, curl and the Laplacian in general orthogonal curvilinear co-ordinate systems Divergence We may now use the form of the gradient to derive the divergence ∇ · A changes with coordinates, so eq wej need to differentiate it too! Before going any further we need to work out ∇ · e qj Back to our definition of the gradient, acting on : qj 21 Our unit vectors form a right-handed set, so But the divergence of the left-hand-side of this equation is zero: I can repeat this argument for cyclic permutations This allows us to rearrange our equation in a nicer way. 22 these have no divergence so can be pulled in front of the ∇ But , so finally we have, 23 Curl Curl works in much the same way. We may use the result we obtained earlier: since, again, the left-hand-side is zero. Remember that so I will pick up contributions for from the terms linked by the arrows. 24 Similarly for the other terms. We can write this as a determinant: 25 Laplacian The laplacian is ∇·∇ = ∇2, so we can fairly easily insert our expression for the gradient into the expression we derived for the divergence: but So, remember both of these derivatives are acting on everything to the right (despite the brackets) 26 Key Point: Apply these expressions to various coordinate systems, including spherical polar coordinates and cylindrical coordinates. Example : Spherical Polar Coordinates: ( , , ) 27 Exercise : Find the gradient, divergence, curl and Laplacian for cylindrical coordinates. Exercise : Using spherical polar coordinates, show that and apply these results to show 28 2. Partial Differential Equations in cylindrical and spherical polar coordinates 2.1 Common Differential Equations Key Point: Write down Laplace’s equation, Poisson’s equation, the Diffusion equation, the Wave equation, the Helmholtz equation and Schrödinger’s equation. We have already seen that physical systems are often described by scalar and vector fields in space and time, that obey particular differential equations involving the operators of the previous section.
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]
  • ' 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]
  • 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]
  • 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]
  • On Sobolev Orthogonality for the Generalized Laguerre Polynomials* Teresa E
    Journal of Approximation Theory AT2973 journal of approximation theory 86, 278285 (1996) article no. 0069 On Sobolev Orthogonality for the Generalized Laguerre Polynomials* Teresa E. Perez and Miguel A. Pin~ar- Departamento de Matematica Aplicada, Universidad de Granada, Granada, Spain Communicated by Walter Van Assche Received December 27, 1994; accepted in revised form September 4, 1995 (:) The orthogonality of the generalized Laguerre polynomials, [Ln (x)]n0,isa well known fact when the parameter : is a real number but not a negative integer. In fact, for &1<:, they are orthogonal on the interval [0, +) with respect to the weight function \(x)=x:e&x, and for :<&1, but not an integer, they are orthogonal with respect to a non-positive definite linear functional. In this work we will show that, for every value of the real parameter :, the generalized Laguerre polynomials are orthogonal with respect to a non-diagonal Sobolev inner product, that is, an inner product involving derivatives. 1996 Academic Press, Inc. 1. INTRODUCTION Classical Laguerre polynomials are the main subject of a very extensive (:) literature. If we denote by [Ln (x)]n0 the sequence of monic Laguerre polynomials, their crucial property is the following orthogonality condition + (:) (:) (:) (:) (:) : &x (Ln , Lm )L = Ln (x) L m (x) x e dx=kn $n, m , (1.1) |0 where m, n # N, kn{0 and the parameter : satisfies the condition &1<: in order to assure the convergence of the integrals. For the monic classical Laguerre polynomials, an explicit representation is well known (see Szego [5, p. 102]): n j (:) n (&1) n+: j Ln (x)=(&1) n! : x , n0, (1.2) j j! \n&j+ =0 a where ( b) is the generalized binomial coefficient a 1(a+1) = .
    [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]
  • Some Identities for the Generalized Laguerre Polynomials
    Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 3388{3396 Research Article Some identities for the generalized Laguerre polynomials Wen-Kai Shaoa, Yuan Heb,∗, Jing Panb aDepartment of Mathematical Teaching and Research, Yibin Vocational & Technical College, 644003 Yibin, Sichuan, P. R. China. bFaculty of Science, Kunming University of Science and Technology, 650500 Kunming, Yunnan, P. R. China. Communicated by R. Saadati Abstract In this paper, we perform a further investigation for the generalized Laguerre polynomials. By applying the generating function methods and Pad´eapproximation techniques, we establish some new identities for the generalized Laguerre polynomials, and give some illustrative special cases as well as immediate consequences of the main results. c 2016 All rights reserved. Keywords: Generalized Laguerre polynomials, Pad´eapproximation, combinatorial identities. 2010 MSC: 11B83, 42C05, 41A21, 05A19. 1. Introduction (α) The generalized Laguerre polynomials Ln (x) associated with non-negative integer n and real number α > −1 are widely used in many problems of mathematical physics and quantum mechanics, for example, in the integration of Helmholtz's equation in paraboloidal coordinates, in the theory of the propagation of electromagnetic oscillations along long lines, etc., as well as in physics in connection with the solution of the second-order linear differential equation: xy00 + (α + 1 − x)y0 + ny = 0: (1.1) These polynomials satisfy some recurrence relations. One very useful, when extracting properties of the wave functions of the hydrogen atom, is the following three-term recurrence relation (see, e.g., [10, 19]): 2n + 1 + α − x n + α L(α) (x) = L(α)(x) − L(α) (x)(n ≥ 1); (1.2) n+1 n + 1 n n + 1 n−1 ∗Corresponding author Email addresses: wksh [email protected] (Wen-Kai Shao), [email protected],[email protected] (Yuan He), [email protected] (Jing Pan) Received 2016-02-28 W.-K.
    [Show full text]
  • Arxiv:1705.01190V1 [Math.CA]
    Noname manuscript No. (will be inserted by the editor) Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions T. M. Dunster · A. Gil · J. Segura Received: date / Accepted: date Abstract Uniform asymptotic expansions involving exponential and Airy (α) functions are obtained for Laguerre polynomials Ln (x), as well as comple- mentary confluent hypergeometric functions. The expansions are valid for n large and α small or large, uniformly for unbounded real and complex values of (α) x. The new expansions extend the range of computability of Ln (x) compared to previous expansions, in particular with respect to higher terms and large values of α. Numerical evidence of their accuracy for real and complex values of x is provided. Keywords Asymptotic expansions Laguerre polynomials Confluent hypergeometric functions Turning point· theory, WKB methods· Numerical computation · · Mathematics Subject Classification (2000) MSC 34E05 33C45 33C15 34E20 33F05 · · · · The authors acknowledge support from Ministerio de Econom´ıa y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE). T. M. Dunster Department of Mathematics and Statistics San Diego State University. 5500 Campanile Drive San Diego, CA, USA. E-mail: [email protected] A. Gil Departamento de Matem´atica Aplicada y CC. de la Computaci´on. ETSI Caminos. Universidad de Cantabria. 39005-Santander, Spain. arXiv:1705.01190v1 [math.CA] 2 May 2017 E-mail: [email protected] J. Segura Departamento de Matem´aticas, Estad´ıstica y Computaci´on. Universidad de Cantabria. 39005-Santander, Spain. E-mail: [email protected] 2 T. M. Dunster et al. 1 Introduction In this paper we shall obtain computable asymptotic expansions for Laguerre (α) polynomials Ln (x), for the case n large and α small or large.
    [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]