Showing papers on "Discrete orthogonal polynomials published in 2013"
TL;DR: In this article, the authors derived new identities related to the Frobenius-Euler polynomials and generalized Carliz's results. And they also gave a relation between the generalized Frobius Euler Polynomial and the generalized Hurwitz-Lerch zeta function at negative integers.
Abstract: The aim of this paper is to derive some new identities related to the Frobenius-Euler polynomials. We also give relation between the generalized Frobenius-Euler polynomials and the generalized Hurwitz-Lerch zeta function at negative integers. Furthermore, our results give generalized Carliz’s results which are associated with Frobenius-Euler polynomials. MSC:05A10, 11B65, 28B99, 11B68.
252 citations
TL;DR: In this article, a class of bilinear differential operators is introduced by assigning specific signs to derivatives, and the corresponding corresponding corresponding linear equations are characterized by Bell polynomials, and two illustrative examples are computed by an algorithm adopting weights of dependent variables.
144 citations
TL;DR: In this article, the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian types were derived using functional equations and differential equations.
Abstract: The first aim of this paper is to construct new generating functions for the generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers. We derive various functional equations and differential equations using these generating functions. The second aim is to provide a novel approach to derive identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, we derive some new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.
124 citations
TL;DR: In this paper, the authors derived explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of arbitrary SU ( N ) group (the HOMFLY polynomials).
92 citations
TL;DR: Every exceptional orthogonal polynomial system is related to a classical system by a Darboux–Crum transformation and a Bochner-type theorem is proved classifying all possible X2-OPSs.
Abstract: Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm–Liouville problems, but without the assumption that an eigenpolynomial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by Carinena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux–Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPSs. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.
88 citations
TL;DR: In this paper, the Witt-type formula for the Changhee numbers and polynomials is considered and some new interesting identities and properties of those polynomial and numbers from the Witt type formula are derived.
Abstract: Recently, Changhee polynomials and numbers are introduced in (6). Some interesting identities and properties of those polynomials are derived from umbral calculus(see (6)). In this paper, we consider Witt-type formula for the Changhee numbers and polynomials and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to special polynomials.
83 citations
TL;DR: In this article, the tangent numbers Tn and polynomials Tn(x) were introduced, and some interesting results and relationships were obtained for mathematics subject classification.
Abstract: In this paper we introduce the tangent numbers Tn and polynomials Tn(x). Some interesting results and relationships are obtained. Mathematics Subject Classification: 11B68, 11S40, 11S80
74 citations
TL;DR: In this article, the authors introduced and investigated several new identities related to a unification and generalization of the three families of generalized Apostol type polynomials, such as the Apostol-Bernoulli polynomial, the Euler polynomorphism, and the Genocchi polynomalization, based on the theory of the Umbral Calculus and the UML.
72 citations
TL;DR: In this article, the authors give operational formulae of the Burchnall type involving complex Hermite polynomials, including Nielsen's identities and Runge addition formula, and give a proof of Nielsen's identity.
Abstract: We give operational formulae of the Burchnall type involving complex Hermite polynomials. Short proofs of some known formulae are given and new results involving these polynomials, including Nielsen's identities and Runge addition formula, are derived.
66 citations
TL;DR: In this article, a class of generalized Bernoulli polynomials and Euler polynomial based on the q-integers is introduced and investigated, and a q-analogues of well-known formulas are derived.
Abstract: The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava-Pinter addition theorem is obtained. We give new identities involving q-Bernstein polynomials.
61 citations
TL;DR: Unlike the previous algorithms, the derived recurrence relation depends neither on the degree nor on the azimuthal order of the radial polynomials, which leads to a reduction in the computational complexity.
Abstract: In optics, Zernike polynomials are widely used in testing, wavefront sensing, and aberration theory. This unique set of radial polynomials is orthogonal over the unit circle and finite on its boundary. This Letter presents a recursive formula to compute Zernike radial polynomials using a relationship between radial polynomials and Chebyshev polynomials of the second kind. Unlike the previous algorithms, the derived recurrence relation depends neither on the degree nor on the azimuthal order of the radial polynomials. This leads to a reduction in the computational complexity.
TL;DR: In this article, asymptotics for Christoffel functions associated with multivariate orthogonal polynomials were established for measures on the ball and simplex under far more general conditions than previously known.
Abstract: We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain. In particular, this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.
TL;DR: In this paper, cyclotomy was used to construct permutation polynomials over finite fields, which allowed the authors to generate polynomial-time permutation algorithms in an algorithmic way and unify several previous constructions.
TL;DR: In this article, two new families of general parameters generalized Jacobi polynomials are introduced for solving third and fifth-order differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a dual Petrov-Galerkin method.
Abstract: Two new families of general parameters generalized Jacobi polynomials are introduced. Some efficient and accurate algorithms based on these families are developed and implemented for solving third- and fifth-order differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a dual Petrov-Galerkin method. The use of general parameters generalized Jacobi polynomials leads to simplified analysis, more precise error estimates and well conditioned algorithms. The method leads to linear systems with specially structured matrices that can be efficiently inverted. Numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented. Keywords: Dual-Petrov-Galerkin method, general parameters generalized Jacobi polynomials, nonhomogeneous Dirichlet conditions Quaestiones Mathematicae 36(2013), 15–38
TL;DR: In this paper, the wavefunctions and scalar products of a one-parameter family of integrable five vertex models are examined and a generalized Cauchy identity satisfied by Grothendieck polynomials and their duals are derived.
Abstract: We examine the wavefunctions and their scalar products of a one-parameter family of integrable five vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on/off-shell wavefunctions of the five vertex models are represented as a certain determinant form. Up to some normalization factors, we find the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of Grothendieck polynomials.
TL;DR: In this paper, a one-parameter family of operators that have the complementary Bannai{ Ito polynomials as eigenfunctions is obtained, which correspond to a q! 1 limit of the Askey{ Wilson polynomial.
Abstract: A one-parameter family of operators that have the complementary Bannai{ Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai{Ito polynomials and also correspond to a q! 1 limit of the Askey{ Wilson polynomials. The eigenvalue equations for the CBI polynomials are found to involve second order Dunkl shift operators with reflections and exhibit quadratic spectra. The algebra associated to the CBI polynomials is given and seen to be a deformation of the Askey{Wilson algebra with an involution. The relation between the CBI polynomials and the recently discovered dual 1 Hahn and para-Krawtchouk polynomials, as well as their relation with the symmetric Hahn polynomials, is also discussed.
TL;DR: A generalized expression of the weighted dual Hahn moment invariants up to any order and for any value of their parameters, as a linear combination of monomials is introduced.
TL;DR: In this paper, some known results for locating the roots of polynomials are extended to the case of matrix polynomial matrices, and these extensions are applied to determine effective initial approximations for the numerical computation of the eigenvalues of the matrix poynomials by means of simultaneous iterations, like the Ehrlich-Aberth method.
Abstract: Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by Pellet [Bull. Sci. Math. (2), 5 (1881), pp. 393--395], some results from Bini [Numer. Algorithms, 13 (1996), pp. 179--200] based on the Newton polygon technique, and recent results from Gaubert and Sharify (see, in particular, [Tropical scaling of polynomial matrices, Lecture Notes in Control and Inform. Sci. 389, Springer, Berlin, 2009, pp. 291--303] and [Sharify, Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues, Ph.D. thesis, Ecole Polytechnique, Paris, 2011]). These extensions are applied to determine effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich--Aberth method. Numerical experiments that show the computational advantage of these results are presented.
TL;DR: In this article, a further investigation for the Carlitz's q-Bernoulli numbers and polynomials was performed, and several symmetric identities for these numbers were established by applying elementary methods and techniques.
Abstract: In this paper, a further investigation for the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials is performed, and several symmetric identities for these numbers and polynomials are established by applying elementary methods and techniques. It turns out that various known results are derived as special cases.
TL;DR: In this article, a quasi-exactly solvable Rabi model is investigated within the framework of the Bargmann Hilbert space of analytic functions, and the eigenvalue equation and eigenfunctions are determined in terms of three systems of monic orthogonal polynomials.
Book•
05 Dec 2013
TL;DR: In this paper, a detailed description of the Riemann-Hilbert approach (RH approach) to the asymptotic analysis of both continuous and discrete orthogonal polynomials, and applications to random matrix models as well as to the six-vertex model was provided.
Abstract: This book provides a detailed description of the Riemann-Hilbert approach (RH approach) to the asymptotic analysis of both continuous and discrete orthogonal polynomials, and applications to random matrix models as well as to the six-vertex model. The RH approach was an important ingredient in the proofs of universality in unitary matrix models. This book gives an introduction to the unitary matrix models and discusses bulk and edge universality. The six-vertex model is an exactly solvable two-dimensional model in statistical physics, and thanks to the Izergin-Korepin formula for the model with domain wall boundary conditions, its partition function matches that of a unitary matrix model with nonpolynomial interaction. The authors introduce in this book the six-vertex model and include a proof of the Izergin-Korepin formula. Using the RH approach, they explicitly calculate the leading and subleading terms in the thermodynamic asymptotic behavior of the partition function of the six-vertex model with domain wall boundary conditions in all the three phases: disordered, ferroelectric, and antiferroelectric.
TL;DR: In this article, the zeros of a family of hypergeometric para-orthogonal polynomials is studied and a monotonicity result for the Meixner-Pollaczek polynomial is proved.
Abstract: Para-orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para-orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para-orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner-Pollaczek polynomials is proved.
TL;DR: A characterization for a point w(|w|=1) to be a pure point of the measure involved is obtained and a characterization for orthogonal polynomials on the unit circle in terms of the two sequences {c"n} and {d"n}.
TL;DR: A new approach that permits the fast computation of Hahn's discrete orthogonal moments is presented and a new set of H Kahn's invariant moments under the translation, the scaling, and the rotation of the image is proposed.
Abstract: The discrete orthogonal moments are powerful descriptors for image analysis and pattern recognition. However, the computation of these moments is a time consuming procedure. To solve this problem, a new approach that permits the fast computation of Hahn’s discrete orthogonal moments is presented in this paper. The proposed method is based, on the one hand, on the computation of Hahn’s discrete orthogonal polynomials using the recurrence relation with respect to the variable x instead of the order n and the symmetry property of Hahn’s polynomials and, on the other hand, on the application of an innovative image representation where the image is described by a number of homogenous rectangular blocks instead of individual pixels. The paper also proposes a new set of Hahn’s invariant moments under the translation, the scaling, and the rotation of the image. This set of invariant moments is computed as a linear combination of invariant geometric moments from a finite number of image intensity slices. Several experiments are performed to validate the effectiveness of our descriptors in terms of the acceleration of time computation, the reconstruction of the image, the invariability, and the classification. The performance of Hahn’s moment invariants used as pattern features for a pattern classification application is compared with Hu [IRE Trans. Inform. Theory8, 179 (1962)] and Krawchouk [IEEE Trans. Image Process.12, 1367 (2003)] moment invariants.
TL;DR: In this article, the concept of D-operators associated to a sequence of polynomials (p"n)"n) and an algebra A of operators acting in the linear space of poynomials was introduced.
TL;DR: In this paper, the spectrum and Krull dimension of the ring Int D (A ) of integer-valued polynomials on A were determined for Noetherian one-dimensional D.
TL;DR: It is presented necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses, and approaches for determining suitable rank- 1 lattices using a component-by-component algorithm are suggested.
Abstract: With given Fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a one-dimensional discrete Fourier transform. In many applications one is also interested in the reconstruction of the Fourier coefficients from samples in the spatial domain. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses. In addition, we suggest approaches for determining suitable rank-1 lattices using a component-by-component algorithm. We present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100.
TL;DR: In this paper, the matrix-valued orthogonal polynomials and the corresponding weight function are studied and the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomsials.
Abstract: In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the P_n's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials P_n. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2)\times SU(2).
TL;DR: In this paper, generalized Fibonacci polynomials were presented and generalized Binet's formula and generating function was used to derive the identities. The proofs of the main theorems are based on simple algebra and give several interesting properties involving them.
Abstract: In this study, we present generalized Fibonacci polynomials We have used their Binet’s formula and generating function to derive the identities The proofs of the main theorems are based on special functions, simple algebra and give several interesting properties involving them
TL;DR: Using Dickson's classification, a family of permutation polynomials from Dicksonʼs list provides counterexamples to a published conjecture of Mullen, and is determined to be the complete list of degree 6 orthomorphism polynmials.