DOCSLIB.ORG

Higher Spherical Polynomials T. Ibukiyama and D. Zagier Contents Introduction 1 Chapter I. General Theory 5 §1. the Generalized

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Higher Spherical Polynomials T. Ibukiyama and D. Zagier Contents Introduction 1 Chapter I. General Theory 5 §1. the Generalized Higher Spherical Polynomials T. Ibukiyama and D. Zagier Contents Introduction 1 Chapter I. General Theory 5 1. The generalized Legendre differential equation 5 § 2. Dimension formula and decomposition theorem 8 § 3. Scalar product and orthogonality 13 § 4. The Lie algebra of mixed Laplacians 19 § 5. The two canonical bases of a(d) 24 § P 6. Higher spherical polynomials and invariant harmonic polynomials 30 § Chapter II. Explicit Constructions 35 7. Construction of invariant harmonic polynomials for d = 4 35 § 8. Higher spherical polynomials for n = 3 38 § A. Generating function 38 B. Ratios of the various basis elements 41 C. Recursion relations 43 D. Explicit formulas for the coefficients of Pνν,d 45 E.Inhomogeneouscoordinatesandangularcoordinates 48 9. A universal generating function for the descending basis 50 § A. Existence of the power series G(n) 51 B. Proofs of the closed formula and commutation relation for 56 Lp C. Special cases and explicit formulas 59 10. Algebraic generating functions for the case n = 4 67 § A. Even values of d : experimental results 67 B. Algebraicity and non-algebraicity proofs for G(4) when s is integral 69 C. Properties of the generating function for fixed integral values of s 74 D. Generating function with respect to s 78 E. Odd values of d 80 F. Final remarks 82 Tables of higher spherical polynomials 83 Table 1. Spherical polynomials for n = 3 83 Table 2. Spherical polynomials and Gram matrices for n = 4 and n = 5 85 Table 3. Determinants of Gram matrices 92 List of Notations 95 Bibliography 96 0 HIGHER SPHERICAL POLYNOMIALS Tomoyoshi Ibukiyama and Don Zagier Abstract. In this paper, we study a new class of special functions. Specifically, we study a vector (n) space P (d) (n ∈ N, d ∈ C) of polynomials P (T ) in n × n symmetric matrices T = (tij ). For integral d ≥ n these are the polynomials on (Rd)n that are invariant with respect to the diagonal action of O(d) and harmonic on each Rd, and for general d they are the polynomial solutions of a certain system of differential equations. For n = 2 these are the classical Legendre and Gegenbauer polynomials, but for n > 2 they are new and are interesting for several reasons, including an application to the theory of Siegel modular forms and the fact that for n = 3 the associated system of differential equations is holonomic. When n = 3, each homogeneous component of P(n)(d) is one-dimensional, so there is a canonical basis. Even here the structure of the polynomials turns out to be very subtle. For n > 3 there is no obvious basis. We construct two canonical bases, dual to one another, and a generating function (generalizing the classical one for n = 2, and algebraic if n = 3 but not in general) for one of them. We also provide tables that illustrate some of the idiosyncracies of the theory. Introduction d For two natural numbers n and d and a scalar product ( , ) on R , we have a map βn from (Rd)n to the space of real symmetric n n matrices, given by Sn × βn : (x1,... ,xn) T =(tij)1 i,j n , tij =(xi,xj) . (1) 7→ ≤ ≤ d n Thus any polynomial P on n defines a polynomial P = P βn on (R ) . If d n, then we (n) S ◦ ≥ denote by (d) the space of polynomials P on n for which P is harmonic with respect to P S n xi for each i = 1,... ,n. For each multidegree a = (ae1,... ,an) (Z 0) we denote by a(d) ≥ (n) ∈ P the subspace of P (d) for which P is a homogeneous of degreee ai with respect to xi for each i = 1,... ,n.∈ (Here P the superscript “(n)” can be omitted since n is just the length of a.) Later in the paper we will generalize thee definitions of both (n)(d) and a(d) to arbitrary P P complex values of d. The elements of (n)(d) will be called higher spherical polynomials. There are several motivations for studying them:P (i) If n = 2, then (d) is one-dimensional if a = a (it is 0 otherwise) and is then P(a1,a2) 1 2 spanned by a polynomial in t11t22 and t12 which is the homogeneous version of one of the classi- cal families of orthogonal polynomials (Legendre polynomials if d = 3, Chebyshev polynomials if d =2 or d = 4, and Gegenbauer polynomial for d arbitrary). These classical polynomials occur in many places in mathematics and mathematical physics and have many nice properties: differ- ential equations, orthogonality properties, recursions, generating functions, closed formulas, etc. It is natural to try to generalize these. (ii) The polynomials in (n)(d) have an application in the theory of Siegel modular forms. This is described in detail inP [7] (similar differential operators are given in [2] and [4]) and will 1 not be pursued in this paper, but was our original reason for studying this particular space of polynomials. Briefly, the connection is as follows. If P a(d) (d n) and if ∈P ≥ F (Z)= c(T ) e2πi Tr(T Z) (Z H = Siegel upper half-space) ∈ n XT is a holomorphic Siegel modular form of weight d/2, then the function 1+ δij ∂ 2πi(t11z1+...+tnnzn) P F (Z) = c(T ) P (T ) e (z1,... ,zn H1) (2) 2 ∂zij Hn ∈ 1 T X is an elliptic modular form of weight d/2+ ai in zi for each i = 1,... ,n. (By considering images n g(H ) Hn, where g is a suitable element of Sp2n(R), one also gets maps from Siegel modular forms⊂ of degree n to Hilbert modular forms on totally real number fields of degree n, or to products of Hilbert modular forms on several totally real fields with degrees adding up to n.) We can understand why this should be true by taking F to be a Siegel theta series, say F (Z)= e2πi Tr( βn(x1,... ,xn) Z) n (x ,... ,xn) L 1 X ∈ where L Rd is a lattice on which the quadratic form (x,x) takes rational values. Then the function defined⊂ by (2) is the series (x1,x1) (xn,xn) 2π√ 1 zi P (x ,... ,x ) q ...q (q = e − ) ··· 1 n 1 n i x L xn L X1∈ X∈ and for P a(d) this ise a modular form of weight d/2 + ai in zi by the classical Hecke- Schoeneberg∈ theory P of theta series with harmonic polynomial coefficients. A deeper direct application of our theory to critical values of triple L function is found in [10] (see also [5]) and another direct application is given in [3]. There are various other theories of differential operators and applications of a similar kind (for example, see [8], [9], [11]). (iii) The spaces a(d) can be looked at from the point of view of representation theory. P Specifically, the map P P gives an isomorphism between a(d) and the space of “invariant harmonic polynomials,”7→ i.e., polynomials in n variables x PRd which are harmonic and ho- i ∈ mogeneous (of degree ai) ine each xi and invariant under the simultaneous action of O(d), the orthogonal group of the scalar product ( , ), on all xi; this space can be analyzed in terms of the classical representation theory of O(d). However, apart from a brief mention of this in connection with the dimension formula proved in 2, we will not use this interpretation, but § will work directly with the functions P on n, since this is more elementary, gives more explicit results, and is applicable in greater generality.S (iv) If n = 3, then the dimension of a(d) is always 0 or 1, so that the higher spherical polynomials are particularly canonical in thisP case. The higher spherical polynomials of varying multidegree give an orthogonal basis of L2 functions in three variables with respect to a certain scalar product, and the coefficients of these polynomials also turn out to be combinatorially very interesting expressions, including as special cases the classical 3j-symbol (or Clebsch-Gordan coefficients) of quantum mechanics. (v) In a sequel to this paper [12], we will show that the system of differential equations defining the higher spherical polynomials (generalized Legendre differential equations) in the case n = 3 is equivalent to an integrable Pfaffian system of rank 8, so that there are precisely eight linearly independent solutions, generalizing the Legendre functions of the first and second kinds in the classical case n = 2. In the general theory of differential equations, such systems are very rare: systems of linear partial differential equations with polynomial coefficients generally have a solution space which is either zero- or infinite-dimensional, and it is non-trivial to construct Pfaffian systems satisfying the necessary integrability conditions. The system arising from the study of higher spherical polynomials is therefore also interesting from this point of view. 2 The contents of the paper are as follows. Chapter I describes the general theory, for arbitrary n and d. In the first two sections, which are basic for everything that follows, we write down the differential equations defining the space (d) (now for any d C), prove a formula for Pn ∈ the dimensions of its homogeneous pieces a(d), and construct a specific basis P M (“monomial P ν basis”) characterized by the property that its restriction to the subspace t11 = = tnn = 0 consists of monomials. We also give a brief description of the behavior of (d) at certain··· special Pn values of d (in 2Z 0) where its dimension changes.
Load more

Recommended publications
  • Ladder Operators for Lam\'E Spheroconal Harmonic Polynomials
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 074, 16 pages Ladder Operators for Lam´eSpheroconal Harmonic Polynomials? Ricardo MENDEZ-FRAGOSO´ yz and Eugenio LEY-KOO z y Facultad de Ciencias, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: [email protected] URL: http://sistemas.fciencias.unam.mx/rich/ z Instituto de F´ısica, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: eleykoo@fisica.unam.mx Received July 31, 2012, in final form October 09, 2012; Published online October 17, 2012 http://dx.doi.org/10.3842/SIGMA.2012.074 Abstract. Three sets of ladder operators in spheroconal coordinates and their respective actions on Lam´espheroconal harmonic polynomials are presented in this article. The poly- nomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lam´epolynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers n1 and n2 counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components L^x, L^y, L^z of the angular momentum connect pairs of the four species of poly- nomials of a chosen kind and angular momentum. The third set of operators, the cartesian componentsp ^x,p ^y,p ^z of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities.
    [Show full text]
  • Arxiv:1406.1948V2 [Math.GM] 18 Dec 2014 Hr = ∆ Where H Ouino 1 Is (1) of Solution the 1
    Solution of Polynomial Equations with Nested Radicals Nikos Bagis Stenimahou 5 Edessa Pellas 58200, Greece [email protected] Abstract In this article we present solutions of arbitrary polynomial equations in nested periodic radicals Keywords: Nested Radicals; Quintic; Sextic; Septic; Equations; Polynomi- als; Solutions; Higher functions; 1 A general type of equations Consider the following general type of equations ax2µ + bxµ + c = xν . (1) Then (1) can be written in the form b 2 ∆2 a xµ + = xν , (2) 2a − 4a where ∆ = √b2 4ac. Hence − 2 µ b ∆ 1 ν x = − + 2 + x (3) 2a r4a a Set now √d x = x1/d, d Q 0 , then ∈ + −{ } 2 µ b ∆ 1 x = − + + xν (4) s 2a 4a2 a arXiv:1406.1948v2 [math.GM] 18 Dec 2014 r hence µ/ν 2 ν b ∆ 1 ν x = − + 2 + x (5) s 2a r4a a Theorem 1. The solution of (1) is µ v 2 µ/ν 2 2 u b v ∆ 1 b ∆ 1 µ/ν b ∆ x = u− + u + v− + v + − + + ... (6) u 2a u4a2 a u 2a u4a2 a s 2a 4a2 u u u u r u u u t t t t 1 Proof. Use relation (4) and repeat it infinite times. Corollary 1. The equation ax6 + bx5 + cx4 + dx3 + ex2 + fx + g = 0 (7) is always solvable with nested radicals. Proof. We know (see [1]) that all sextic equations, of the most general form (7) are equivalent by means of a Tchirnhausen transform y = kx4 + lx3 + mx2 + nx + s (8) to the form 6 2 y + e1y + f1y + g1 = 0 (9) But the last equation is of the form (1) with µ = 1 and ν = 6 and have solution 2 1/6 2 2 f v ∆ 1 f ∆ 1 1/6 f ∆ y = 1 + u 1 v 1 + 1 1 + 1 + ..
    [Show full text]
  • ALGEBRAIC GEOMETRY (1) Compute the Points of Intersection
    ALGEBRAIC GEOMETRY HOMEWORK 5 (1) Compute the points of intersection of the circle x2 +y2 +x−y = 0 with the lemniscate (x2 +y2)2 = x2 −y2 using resultants (can you compute the affine points of intersections directly?), and compute the multiplicities. Explain which coordinate system you choose, how the equations look like, what the resultant is, and what its factors are. (2) Here you will learn how to solve cubics using resultants. Assume you want to solve the cubic equation x3 + ax2 + bx + c = 0. (a) Show that the substitution y = x + a/3 transforms the cubic into y3 + By + C = 0. (b) Now assume that we have to solve x3 + bx + c = 0. Using linear re- lations y = rx ∗ b will not make the linear term disappear (without bringing back the quadratic term). Thus we try to put y = x2 + cx + d and then eliminate x using resultants: type in p = polresultant(x^3+b*x+c,y-x^2-d*x-e,x) polcoeff(p,2,y) polcoeff(p,1,y) (you can find out what polcoeff is doing by typing ?polcoeff). Now we know that we can keep the quadratic term out of the equation by choosing e = 2b/3. With this value of e go back and recompute the resultant; this will now be a cubic in y without quadratic term. The coefficient of the linear term can be made to disappear by solving a quadratic equation (which is called the resolvent of the original cubic). (c) Thus you can solve the cubic by solving the resolvent, using the trans- formations above to turn the cubic into a pure cubic y3 + c, which in turn is easy to solve.
    [Show full text]
  • An Algebraic Approach to Harmonic Polynomials on S3
    AN ALGEBRAIC APPROACH TO HARMONIC POLYNOMIALS ON S3 Kevin Mandira Limanta A thesis in fulfillment of the requirements for the degree of Master of Science (by Research) SCHOOL OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCE UNIVERSITY OF NEW SOUTH WALES June 2017 ORIGINALITY STATEMENT 'I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.' Signed .... Date Show me your ways, Lord, teach me your paths. Guide me in your truth and teach me, for you are God my Savior, and my hope is in you all day long. { Psalm 25:4-5 { i This is dedicated for you, Papa. ii Acknowledgement This thesis is the the result of my two years research in the School of Mathematics and Statistics, University of New South Wales. I learned quite a number of life lessons throughout my entire study here, for which I am very grateful of.
    [Show full text]
  • Spherical Harmonics and Homogeneous Har- Monic Polynomials
    SPHERICAL HARMONICS AND HOMOGENEOUS HAR- MONIC POLYNOMIALS 1. The spherical Laplacean. Denote by S ½ R3 the unit sphere. For a function f(!) de¯ned on S, let f~ denote its extension to an open neighborhood N of S, constant along normals to S (i.e., constant along rays from the origin). We say f 2 C2(S) if f~ is a C2 function in N , and for such functions de¯ne a di®erential operator ¢S by: ¢Sf := ¢f;~ where ¢ on the right-hand side is the usual Laplace operator in R3. With a little work (omitted here) one may derive the expression for ¢ in polar coordinates (r; !) in R2 (r > 0;! 2 S): 2 1 ¢u = u + u + ¢ u: rr r r r2 S (Here ¢Su(r; !) is the operator ¢S acting on the function u(r; :) in S, for each ¯xed r.) A homogeneous polynomial of degree n ¸ 0 in three variables (x; y; z) is a linear combination of `monomials of degree n': d1 d2 d3 x y z ; di ¸ 0; d1 + d2 + d3 = n: This de¯nes a vector space (over R) denoted Pn. A simple combinatorial argument (involving balls and separators, as most of them do), seen in class, yields the dimension: 1 d := dim(P ) = (n + 1)(n + 2): n n 2 Writing a polynomial p 2 Pn in polar coordinates, we necessarily have: n p(r; !) = r f(!); f = pjS; where f is the restriction of p to S. This is an injective linear map p 7! f, but the functions on S so obtained are rather special (a dn-dimensional subspace of the in¯nite-dimensional space C(S) of continuous functions-let's call it Pn(S)) We are interested in the subspace Hn ½ Pn of homogeneous harmonic polynomials of degree n (¢p = 0).
    [Show full text]
  • On the Solution to Nonic Equations
    On the Solution to Nonic Equations By Dr. Raghavendra G. Kulkarni Abstract. This paper presents a novel decomposition tech- nique in which a given nonic equation is decomposed into quartic and quintic polynomials as factors, eventually lead- ing to its solution in radicals. The conditions to be satisfied by the roots and the coefficients of such a solvable nonic equation are derived. Introduction It is well known that the general polynomial equations of degree five and above cannot be solved in radicals, and have to be solved algebraically using symbolic coefficients, such as the use of Bring radicals for solving the general quintic equation, and the use of Kampe de Feriet functions to solve the general sextic equation [1, 2]. There is hardly any literature on the analytical solution to the polynomial equations beyond sixth degree. The solution to certain solvable octic equations is described in a recent paper [4]. This paper presents a method of determining all nine roots of a nonic equation (ninth-degree polynomial equation) in radicals. First, the given nonic equation is decomposed into two factors in a novel fashion. One of these factors is a fourth-degree polynomial, while the other one is a fifth-degree polynomial. When the fourth- degree polynomial factor is equated to zero, the four roots of the given nonic equation are extracted by solving the resultant quar- tic equation using the well-known Ferrari method or by a method given in a recent paper [3]. However, when the fifth-degree polyno- mial factor is equated to zero, the resultant quintic equation is also required to be solved in radicals, since the aim of this paper is to solve the nonic equation completely in radicals.
    [Show full text]
  • Harmonic Analysis on the Proper Velocity Gyrogroup ∗ 1 Introduction
    Harmonic Analysis on the Proper Velocity gyrogroup ∗ Milton Ferreira School of Technology and Management, Polytechnic Institute of Leiria, Portugal 2411-901 Leiria, Portugal. Email: [email protected] and Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193 Aveiro, Portugal. Email [email protected] Abstract In this paper we study harmonic analysis on the Proper Velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativis- tic addition of proper velocities in special relativity and it is related with the hy- perboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter z and on the radius t of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherel's Theorem. In the limit of large t; t ! +1; the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on Rn; thus unifying hyperbolic and Euclidean harmonic analysis. Keywords: PV gyrogroup, Laplace Beltrami operator, Eigenfunctions, Generalized Helgason- Fourier transform, Plancherel's Theorem. 1 Introduction Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves called harmonics. It investigates and gen- eralizes the notions of Fourier series and Fourier transforms. In the past two centuries, it has become a vast subject with applications in diverse areas as signal processing, quantum mechanics, and neuroscience (see [18] for an overview).
    [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]
  • Ladder Operators for Lamé Spheroconal Harmonic
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 074, 16 pages Ladder Operators for Lam´eSpheroconal Harmonic Polynomials? Ricardo MENDEZ-FRAGOSO´ yz and Eugenio LEY-KOO z y Facultad de Ciencias, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: [email protected] URL: http://sistemas.fciencias.unam.mx/rich/ z Instituto de F´ısica, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: eleykoo@fisica.unam.mx Received July 31, 2012, in final form October 09, 2012; Published online October 17, 2012 http://dx.doi.org/10.3842/SIGMA.2012.074 Abstract. Three sets of ladder operators in spheroconal coordinates and their respective actions on Lam´espheroconal harmonic polynomials are presented in this article. The poly- nomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lam´epolynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers n1 and n2 counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components L^x, L^y, L^z of the angular momentum connect pairs of the four species of poly- nomials of a chosen kind and angular momentum. The third set of operators, the cartesian componentsp ^x,p ^y,p ^z of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities.
    [Show full text]
  • On Sieved Orthogonal Polynomials Ii: Random Walk Polynomials
    Can. J. Math., Vol. XXXVIII, No. 2, 1986, pp. 397-415 ON SIEVED ORTHOGONAL POLYNOMIALS II: RANDOM WALK POLYNOMIALS JAIRO CHARRIS AND MOURAD E. H. ISMAIL 1. Introduction. A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities (1.1) pmn(t) = Pr{X(t) = n\X(0) = m) satisfy Pmt + 0(/) n = m + 1 (1.2) Pm„(t) = 8mt + 0(0 n = m - 1 1 — 08m + Sm)t + o(0 n = m, as / —> 0. Here we assume /?n > 0, 8n + 1 > 0, « = 0, 1,. , but ô0 ^ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x) } generated by Ôo(*) = 1> Gi(*) = (A) + «b " ^Vi8o, « > 0. In this case there exists a positive measure da supported on [0, oo) such that oo Qn(x)Qm(x)da(x) = 8mn/iTn, m, n = 0, 1,. /; o holds where •un = /yj,... p„-l/{sls2... 8„), « > o, «o = l. The second set is the set of random walk polynomials. They arise when one studies a random walk on the state space. The random walk polynomials {Rn(x) } satisfy the recursion (1.3) xRn(x) = BnRn + x(x) + AA-i(*)> n > 0 and the initial conditions Received October 3, 1984 and in revised form February 14, 1985. This research was partially supported by NSF Grant MCS 8313931, Arizona State University and the National University of Colombia. 397 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • 04. Rukhin-Final5a-116-1.Qxd
    Volume 116, Number 1, January-February 2011 Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 116, 539-556 (2011)] Maximum Likelihood and Restricted Likelihood Solutions in Multiple-Method Studies Volume 116 Number 1 January-February 2011 Andrew L. Rukhin A formulation of the problem of considered to be known an upper bound combining data from several sources is on the between-method variance is National Institute of Standards discussed in terms of random effects obtained. The relationship between and Technology, models. The unknown measurement likelihood equations and moment-type Gaithersburg, MD 20899-8980 precision is assumed not to be the same equations is also discussed. for all methods. We investigate maximum likelihood solutions in this model. By representing the likelihood equations as Key words: DerSimonian-Laird simultaneous polynomial equations, estimator; Groebner basis; [email protected] the exact form of the Groebner basis for heteroscedasticity; interlaboratory studies; their stationary points is derived when iteration scheme; Mandel-Paule algorithm; there are two methods. A parametrization meta-analysis; parametrized solutions; of these solutions which allows their polynomial equations; random effects comparison is suggested. A numerical model. method for solving likelihood equations is outlined, and an alternative to the maximum likelihood method, the restricted Accepted: August 20, 2010 maximum likelihood, is studied. In the situation when methods variances are Available online: http://www.nist.gov/jres 1. Introduction: Meta Analysis and difficult when the unknown measurement precision Interlaboratory Studies varies among methods whose summary results may not seem to conform to the same measured property.
    [Show full text]
  • Moments of Classical Orthogonal Polynomials
    Moments of Classical Orthogonal Polynomials zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr.rer.nat) im Fachbereich Mathematik der Universität Kassel By Patrick Njionou Sadjang ????? Ph.D thesis co-supervised by: Prof. Dr. Wolfram Koepf University of Kassel, Germany and Prof. Dr. Mama Foupouagnigni University of Yaounde I, Cameroon October 2013 Tag der mündlichen Prüfung 21. Oktober 2013 Erstgutachter Prof. Dr. Wolfram Koepf Universität Kassel Zweitgutachter Prof. Dr. Mama Foupouagnigni University of Yaounde I Abstract The aim of this work is to find simple formulas for the moments mn for all families of classical orthogonal polynomials listed in the book by Koekoek, Lesky and Swarttouw [30]. The generating functions or exponential generating functions for those moments are given. To my dear parents Acknowledgments Foremost, I would like to express my sincere gratitude to my advisors Prof. Dr. Wolfram Koepf and Prof. Dr. Mama Foupouagnigni for the continuous support of my Ph.D study and research, for their patience, motivation, enthusiasm, and immense knowledge. Their guidance helped me in all the time of research and writing of this thesis. I could not have imagined having better advisors and mentors for my Ph.D study. I am grateful to Prof. Dr. Mama Foupouagnigni for enlightening me the first glance of re- search. My sincere thanks also go to Prof. Dr. Wolfram Koepf for offering me the opportunity to visit the University of Kassel where part of this work has been written. I acknowledge the financial supports of the DAAD via the STIBET fellowship which en- abled me to visit the Institute of Mathematics of the University of Kassel.
    [Show full text]