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Orthogonal polynomials for exponential weights x2ρe−2Q(x) on [0,d
 J. Approx. Theory
"... xi + 476 pp With each of a large class of positive measures µ on the real line it is possible to associate a sequence {pn(x)} of orthogonal polynomials with the property that pm(x)pn(x)dµ(x) = 0, m ̸ = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xpn(x) = anPn+1(x) + ..."
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Cited by 66 (17 self)
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xi + 476 pp With each of a large class of positive measures µ on the real line it is possible to associate a sequence {pn(x)} of orthogonal polynomials with the property that pm(x)pn(x)dµ(x) = 0, m ̸ = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xpn(x) = anPn+1(x) + bnPn(x) + an−1Pn−1(x) (2) Conversely, for suitable starting values and coefficient sequences {an}, {bn}, the recurrence relation (2) generates a sequence of polynomials satisfying (1) for some measure µ. The polynomials have their zeros within the interval of support of the measure. Examples date from the 19th century. The Jacobi polynomials P (α,β) n (x) are orthogonal with respect to µ with support [−1, 1], where dµ = (1 − x) α+1 (1 +
Tridiagonal pairs and the quantum affine algebra . . .
, 2003
"... Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A,A ∗ denote a tridiagonal pair on V. Let θ0,θ1,...,θd (resp. θ ∗ 0,θ ∗ 1,...,θ ∗ d) denote a standard ordering of ..."
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Cited by 58 (27 self)
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Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A,A ∗ denote a tridiagonal pair on V. Let θ0,θ1,...,θd (resp. θ ∗ 0,θ ∗ 1,...,θ ∗ d) denote a standard ordering of the eigenvalues of A (resp. A ∗). We assume there exist nonzero scalars a,a ∗ in K such that θi = aq 2i−d and θ ∗ i = a ∗ q d−2i for 0 ≤ i ≤ d. We display two irreducible Uq ( ̂ sl2)module structures on V and discuss how these are related to the actions of A and A∗.
Advanced Determinant Calculus
, 1999
"... The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have ..."
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Cited by 58 (0 self)
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The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.
The Asymptotic Zero Distribution of Orthogonal Polynomials With Varying Recurrence Coefficients
, 1999
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Leonard pairs and the AskeyWilson relations
 Department of Computational Science Faculty of Science Kanazawa University Kakumamachi Kanazawa
"... Let V denote a vector space with finite positive dimension, and let (A,A ∗) denote a Leonard pair on V. As is known, the linear transformations A, A ∗ satisfy the AskeyWilson relations A 2 A ∗ − βAA ∗ A + A ∗ A 2 − γ (AA ∗ +A ∗ A) − ̺ A ∗ = γ ∗ A 2 + ωA + η I, A ∗2 A − βA ∗ AA ∗ + AA ∗2 − γ ∗ ( ..."
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Cited by 47 (22 self)
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Let V denote a vector space with finite positive dimension, and let (A,A ∗) denote a Leonard pair on V. As is known, the linear transformations A, A ∗ satisfy the AskeyWilson relations A 2 A ∗ − βAA ∗ A + A ∗ A 2 − γ (AA ∗ +A ∗ A) − ̺ A ∗ = γ ∗ A 2 + ωA + η I, A ∗2 A − βA ∗ AA ∗ + AA ∗2 − γ ∗ (A ∗ A+AA ∗ ) − ̺ ∗ A = γA ∗2 + ωA ∗ + η ∗ I, for some scalars β, γ, γ ∗ , ̺, ̺ ∗ , ω,η, η ∗. The scalar sequence is unique if the dimension of V is at least 4. If c, c ∗ , t, t ∗ are scalars and t, t ∗ are not zero, then (tA+c, t ∗ A ∗ +c ∗ ) is a Leonard pair on V as well. These affine transformations can be used to bring the Leonard pair or its AskeyWilson relations into a convenient form. This paper presents convenient normalizations of Leonard pairs by the affine transformations, and exhibits explicit AskeyWilson relations satisfied by them. 1
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 45 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Stochastic modeling of flow structure interactions using generalized polynomial chaos
 JOURNAL OF FLUIDS ENGINEERING
, 2002
"... We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flowstructure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standar ..."
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Cited by 40 (2 self)
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We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flowstructure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach is a generalization of the original polynomial chaos expansion, which was first introduced by N. Wiener (1938) and employs the Hermite polynomials (a subset of the Askey scheme) as the basis in random space. The algorithm is first applied to secondorder oscillators to demonstrate convergence, and subsequently is coupled to incompressible NavierStokes equations. Error bars are obtained, similar to laboratory experiments, for the pressure distribution on the surface of a cylinder subject to vortexinduced vibrations.
Gibbs sampling, exponential families and orthogonal polynomials
 Statistical Sciences
, 2008
"... Abstract. We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical ort ..."
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Cited by 40 (10 self)
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Abstract. We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. Key words and phrases: Gibbs sampler, running time analyses, exponential families, conjugate priors, location families, orthogonal polynomials, singular value decomposition. 1.
An elementary approach to 6jsymbols (classical, quantum, rational, trigonometric, and elliptic)
, 2003
"... Elliptic 6jsymbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6jsymbols (or qRacah polynomials) and Wilson’s biorthogonal 10W9 functions. We give an elementary construction of elliptic 6jsymbols, which i ..."
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Cited by 38 (2 self)
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Elliptic 6jsymbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6jsymbols (or qRacah polynomials) and Wilson’s biorthogonal 10W9 functions. We give an elementary construction of elliptic 6jsymbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6jsymbols in terms of Sklyanin algebra representations.