Showing papers on "Discrete orthogonal polynomials published in 2009"

Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

Abstract: New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same charac- teristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are con- structed. In the linear case, they contain ( + 1)th-degree polynomials with = 0,1,2,..., which are shown to beX1-Laguerre orX1-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of ( + 2)th-degree Laguerre-type polynomials and a single one of ( + 2)th-degree Jacobi-type polynomials with = 0,1,2,... are identified. They are candidates for the still unknown X2-Laguerre and X2-Jacobi exceptional orthogonal polynomials, respectively.

Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤp

Abstract: The objective of the paper is to indicate a symmetry of the multivariate p-adic invariant integral on ℤ p , which leads to a relation between the power sum polynomials and higher-order Euler polynomials.

On k-Fibonacci sequences and polynomials and their derivatives

Abstract: The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven. � 2007 Elsevier Ltd. All rights reserved.

Approximation by Complex Bernstein and Convolution Type Operators

Abstract: Bernstein-Type Operators of One Complex Variable: Auxiliary Results in Complex Analysis Bernstein Polynomials Iterates of Bernstein Polynomials Generalized Voronovskaja Theorems for Bernstein Polynomials Butzer's Linear Combination of Bernstein Polynomials q-Bernstein Polynomials Bernstein-Stancu Polynomials Bernstein-Kantorovich Type Polynomials Favard-Szasz-Mirakjan Operators Baskakov Operators Balazs-Szabados Operators Bibliographical Notes and Open Problems Bernstein-Type Operators of Several Complex Variables: Introduction Bernstein Polynomials Favard-Szasz-Mirakjan Operators Baskakov Operators Bibliographical Notes and Open Problems Complex Convolutions: Linear Polynomial Convolutions Linear Non-Polynomial Convolutions Nonlinear Complex Convolutions Bibliographical Notes and Open Problems Appendices: Related Topics: Bernstein Polynomials of Quaternion Variable Approximation of Vector-Valued Functions Strong Approximation by Complex Taylor Series Bibliographical Notes and Open Problems.

Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals

Abstract: This article is a short elementary review of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of analysis. Some new properties are included, and several analysis-related applications are mentioned. At the end of the paper one application is described in details—certain Fourier integrals involving Γ ( 𝑎 + 𝑖 𝑡 ) and Γ ( 𝑎 + 𝑖 𝑡 ) Γ ( 𝑏 − 𝑖 𝑡 ) are evaluated in terms of Stirling numbers.

Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n

Abstract: Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is differentiable and all of its first partial derivatives are locally Lipshitz continuous. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Ampere measure of the weight function itself on a certain set. A setting of polynomials associated to a given Newton polytope, scaled by k, is also considered. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.

Painlev\'e V and time dependent Jacobi polynomials

Abstract: In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight $w_0(x)$ defined on an interval by $w_0(x)e^{-tx}.$ It is a well-known fact that under such a deformation the recurrence coefficients denoted as $\alpha_n$ and $\beta_n$ evolve in $t$ according to the Toda equations, giving rise to the time dependent orthogonal polynomials, using Sogo's terminology. The resulting "time-dependent" Jacobi polynomials satisfy a linear second order ode. We will show that the coefficients of this ode are intimately related to a particular Painlev\'e V. In addition, we show that the coefficient of $z^{n-1}$ of the monic orthogonal polynomials associated with the "time-dependent" Jacobi weight, satisfies, up to a translation in $t,$ the Jimbo-Miwa $\sigma$-form of the same $P_{V};$ while a recurrence coefficient $\alpha_n(t),$ is up to a translation in $t$ and a linear fractional transformation $P_{V}(\alpha^2/2,-\beta^2/2, 2n+1+\alpha+\beta,-1/2).$ These results are found from combining a pair of non-linear difference equations and a pair of Toda equations. This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painlev\'e equation.

On generalized Fibonacci and Lucas polynomials

Abstract: Let h ( x ) be a polynomial with real coefficients. We introduce h ( x ) -Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h ( x ) -Fibonacci polynomials. We also introduce h ( x ) -Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ( x ) that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

Abstract: Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. The first main result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of a new class of tableaux which we call staircase tableaux. This generalizes our previous work for the ASEP with parameters gamma=delta=0. Using our first result and also results of Uchiyama-Sasamoto-Wadati, we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980's there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.

Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials

Abstract: We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function We also derive the integral representations for the classical Bernoulli and Euler polynomials and related known results

N-fold Supersymmetry and Quasi-solvability Associated with X_2-Laguerre Polynomials

Abstract: We construct a new family of quasi-solvable and N-fold supersymmetric quantum systems where each Hamiltonian preserves an exceptional polynomial subspace of codimension 2. We show that the family includes as a particular case the recently reported rational radial oscillator potential whose eigenfunctions are expressed in terms of the X_2-Laguerre polynomials of the second kind. In addition, we find that the two kinds of the X_2-Laguerre polynomials are ingeniously connected with each other by the N-fold supercharge.

On Some Properties of Horadam Polynomials

Abstract: In this study, we introduce a new generalization of the second order polynomial sequences. Namely, we define the Horadam polynomials sequence. Afterwards, we investigate the some properties of the Horadam polynomials.

A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations

Abstract: This article analyzes the solution of the integrated forms of fourth-order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (−1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ∓½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = −β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two-dimensional version of the approximations is obtained by tensor products of the one-dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well-known condition number O(N8) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to Nd+1 for a d-dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Some Families of Orthogonal Polynomials of a Discrete Variable and their Applications to Graphs and Codes

Abstract: We present some related families of orthogonal polynomials of a discrete variable and survey some of their applications in the study of (distance-regular) graphs and (completely regular) codes. One of the main peculiarities of such orthogonal systems is their non-standard normalization condition, requiring that the square norm of each polynomial must equal its value at a given point of the mesh. For instance, when they are defined from the spectrum of a graph, one of these families is the system of the predistance polynomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polynomials. The applications range from (quasi-spectral) characterizations of distance-regular graphs, walk-regular graphs, local distance-regularity and completely regular codes, to some results on representation theory.

Discrete painlevé equations for recurrence coefficients of semiclassical laguerre polynomials

Abstract: We consider two semiclassical extensions of the Laguerre weight and their associated sets of orthogonal polynomials. These polynomials satisfy a three-term recurrence relation. We show that the coefficients appearing in this relation satisfy discrete Painleve equations.

On Multiple Interpolation Functions of the Nörlund-Type -Euler Polynomials

Abstract: In (2006) and (2009), Kim defined new generating functions of the Genocchi, Norlund-type -Euler polynomials and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type -zeta function. This function interpolates Norlund-type -Euler polynomials at negative integers. We also give some identities related to these polynomials and functions. Furthermore, we give some remarks about approximations of Bernoulli and Euler polynomials.

Computing with Expansions in Gegenbauer Polynomials

Abstract: We develop fast algorithms for computations involving finite expansions in Gegenbauer polynomials. A method is described to convert any finite expansion between different families of Gegenbauer polynomials. For a degree-$n$ expansion the computational cost is $\mathcal{O}(n(\log(1/\varepsilon)+|\alpha-\beta|))$, where $\varepsilon$ is the prescribed accuracy, and $\alpha$ and $\beta$ are the respective Gegenbauer indices. Special cases involving Chebyshev polynomials of first kind are particularly important. In combination with (nonequispaced) discrete cosine transforms, we obtain efficient methods for the evaluation of expansions at prescribed nodes, and for the projection onto a sequence of Gegenbauer polynomials from given function values, respectively.