Showing papers on "Orthogonal polynomials published in 1995"
TL;DR: In this article, the authors review recent progress in 2D gravity coupled to d < 1 conformal matter, based on a representation of discrete gravity in terms of random matrices and discuss the saddle point approximation for these models, including a class of related O(n) matrix models.
1,408 citations
TL;DR: In this paper, the recurrence coefficients of semi-classical orthogonal polynomials related to a weight function w such that w/w is a rational function were shown to be solutions of nonlinear differential equations with respect to a well-chosen parameter.
245 citations
TL;DR: In this article, it was shown that every system of polynomials satisfying some (2 N + 1)-term recurrence relation can be expressed in terms of orthonormal matrix polynomorphisms for which the coefficients are N × N matrices.
194 citations
TL;DR: A flagged Littlewood—Richardson rule, an expansion of a flagged skew Schur function as a nonnegative sum of key polynomials, and two new combinatorial interpretations for these key poynomials are given.
173 citations
TL;DR: In this paper, it was shown that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials, and that the vanishing norm of the norm of a polynomial whose index is greater than a critical value of the energy is the zero of the critical poynomial.
Abstract: This paper shows that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials $\{ P_n\}$. The quantum-mechanical wave function is the generating function for the $P_n (E)$, which are polynomials in the energy $E$. The condition of quasi-exact solvability is reflected in the vanishing of the norm of all polynomials whose index $n$ exceeds a critical value $J$. The zeros of the critical polynomial $P_J(E)$ are the quasi-exact energy eigenvalues of the system.
159 citations
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1 Jan 1995
136 citations
TL;DR: In this paper, the generalization of the Adomian polynomials for a function of p variables and the relation between the adomian and Bell polynomial are discussed.
134 citations
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TL;DR: In this paper, a class of quantum analogues of compact symmetric spaces of classical type is introduced by means of constant solutions to the reflection equations, and their zonal spherical functions are discussed in connection with $q$-orthogonal polynomials.
Abstract: A class of quantum analogues of compact symmetric spaces of classical type is introduced by means of constant solutions to the reflection equations. Their zonal spherical functions are discussed in connection with $q$-orthogonal polynomials.
119 citations
TL;DR: In this article, the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products were investigated and compared with the orthogonality corresponding to the measure μ.
Abstract: We investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products
$$\left\langle {h,{\text{ }}g} \right\rangle = \int h g d\mu + \sum {_{j = 1}^m } \sum {_{i = 0}^{N_j } M_{j,i} h^{(i)} (c_j )} g^{(i)} (c_j )$$
, where μ is a certain type of complex measure on the real line, andc
j are complex numbers in the complement of supp(μ). The Sobolev orthogonal polynomials are compared with the orthogonal polynomials corresponding to the measure μ.
91 citations
TL;DR: In this article, the Wall and Laguerre polynomials are shown to arise as matrix elements of q-exponentials of the generators in a representation of the q-oscillator algebra.
Abstract: Properties of certain q-orthogonal polynomials are connected to the q-oscillator algebra. The Wall and q-Laguerre polynomials are shown to arise as matrix elements of q-exponentials of the generators in a representation of this algebra. A realization is presented where the continuous q-Hermite polynomials form a basis of the representation space. Various identities are interpreted within this model. In particular, the connection formula between the continuous big q-Hermite and the continuous q-Hermite polynomials is thus obtained, and two generating functions for these last polynomials are derived algebraically.
90 citations
TL;DR: In this paper, the Wall and Laguerre polynomials are shown to arise as matrix elements of the generators in a representation of the cosillator algebra, and the connection formula between the continuous big $q$-Hermite polynomorphisms and the continuous $ q$-HERMITIME poynomials is obtained.
Abstract: Properties of certain $q$-orthogonal polynomials are connected to the $q$-oscillator algebra. The Wall and $q$-Laguerre polynomials are shown to arise as matrix elements of $q$-exponentials of the generators in a representation of this algebra. A realization is presented where the continuous $q$-Hermite polynomials form a basis of the representation space. Various identities are interpreted within this model. In particular, the connection formula between the continuous big $q$-Hermite polynomials and the continuous $q$-Hermite polynomials is thus obtained, and two generating functions for these last polynomials are algebraically derived.
TL;DR: In this article, the generalized discrete variable representation (GDVR) was proposed to restore the variational principle of the Fourier method of Kosloff and the DVR of Light by introducing more grid points than spectral basis functions.
Abstract: In developing a pseudospectral transform between a nondirect product basis of spherical harmonics and a direct product grid, Corey and Lemoine [J. Chem. Phys. 97, 4115 (1992)] generalized the Fourier method of Kosloff and the discrete variable representation (DVR) of Light by introducing more grid points than spectral basis functions. Assuming that the potential energy matrix is diagonal on the grid destroys the variational principle in the Fourier and DVR methods. In the present article we (1) demonstrate that the extra grid points in the generalized discrete variable representation (GDVR) act as dealiasing functions that restore the variational principle and make a pseudospectral calculation equivalent to a purely spectral one, (2) describe the general principles for extending the GDVR to other nondirect product spectral bases of orthogonal polynomials, and (3) establish the relation between the GDVR and the least squares method exploited in the pseudospectral electronic structure and adiabatic pseudospectral bound state calculations of Friesner and collaborators.
TL;DR: In this article, background estimation and modelling strategies suitable for evaluating energy-dispersive X-ray spectra by means of non-linear least squares fitting are evaluated and intercompared.
TL;DR: In this paper, the limit behavior of the sequence of Lp-norms of orthogonal polynomials is studied, and asymptotic formulas for entropy integrals containing orthogonality are obtained.
Abstract: The limit behavior of the sequence of Lp-norms of orthogonal polynomials is studied. Orthogonal polynomials on both a finite interval and the entire real line are considered. As a corollary, asymptotic formulas for entropy integrals containing orthogonal polynomials are obtained.Bibliography: 18 titles.
TL;DR: In this paper, rational transfer function models based on orthogonal Forsythe polynomials minimize the condition number of the Jacobian of estimators in a least-squares framework.
Abstract: It is shown that rational transfer function models based on orthogonal Forsythe polynomials minimize the condition number of the Jacobian of estimators in a least-squares framework. As a result, very high order linear time-invariant systems can be identified. The numerical stability of the estimation of the parameters and their derived quantities (zeros, poles, ...) are obtained. Statistical uncertainty bounds are provided. The method is illustrated on a 100th order simulated system and a 120th order measured beam-structure. >
TL;DR: In this paper, a Lie algebraic procedure of discretization of differential equations with isospectral property is proposed using sl2-algebra based approach, (quasi-exactly-solvable finite difference equations are described.
Abstract: A certain representation for the Heisenberg algebra in finite difference operators is established. The Lie algebraic procedure of discretization of differential equations with isospectral property is proposed. Using sl2-algebra based approach, (quasi)-exactly-solvable finite difference equations are described. It is shown that the operators having the Hahn, Charlier and Meissner polynomials as the eigenfunctions are reproduced in the present approach as some particular cases. A discrete version of the classical orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is introduced.
TL;DR: The vibration analysis of shear deformable plates formulated on the basis of first order Mindlin theory is presented in this article, where displacement and rotational functions of the plates are approximated by sets of boundary characteristic orthogonal polynomials.
TL;DR: In this paper, the problem of characterizing all the sequences of orthogonal polynomials in a given system of monic monic polynomorphisms was studied in the case that the given system is a classical one.
TL;DR: In this article, the authors used nonparametric identification techniques to process recorded data of nonlinear structural responses and to represent the constitutive relationship of the structure of hysteretic oscillators.
Abstract: Nonparametric identification techniques are used to process recorded data of nonlinear structural responses and to represent the constitutive relationship of the structure. When hysteretic systems are dealt with, attention must be given to the appropriate subspace of the state variables in which the restoring force can be approximated by a single-valued surface. Nonparametric models are investigated, defined by two different descriptions: the first, in which the restoring force is a function of displacement and velocity, is commonly used; and the second, in which the incremental force is a function of force and velocity is less adopted. The ability of the second variable space to better reproduce the behavior of hysteretic oscillators is shown by analyzing different cases. Meanwhile, approximation of the real restoring function in terms of orthogonal (Chebyshev) polynomials and nonorthogonal polynomials is investigated. Finally, a mixed parametric and nonparametric model that exhibits a very satisfactory behavior in the case of important hardening and viscous damping is presented.
TL;DR: This work gives the representation of Tchebychev co-recursive forms for any complex value of the parameter and various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences.
Abstract: We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S
2(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.
TL;DR: In this article, the authors obtained a set of polynomials which are orthogonal with respect to the classical discrete weight function of the Charlier polynomial at which an extra point mass at x = 0 is added.
TL;DR: In this article, it was shown that yn(X) is irreducible for all but finitely many (possibly 0) positive integers n. The coefficient of xJ in Yn(x) is (2 +j) I-I~=1(2k-1) and hence integral.
Abstract: are all irreducible over the rationals and obtained several results concerning their irreducibility. The statement of this conjecture and his results are described in his book Bessel Polynomials [7]. The author in [4] established that almost all Bessel Polynomials are irreducible. More precisely, if k(t) denotes the number of n<.t for which y,~(x) is reducible, then k(t)=o(t). He later [5] observed that the argument could be strengthened to obtain k(t)<
TL;DR: New upper bounds for code distance and covering radius of designs in arbitrary polynomial metric spaces are presented and it is proved that this bound is attained for all tight 2k-design.
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TL;DR: In this article, a polynomial calculus compatible with polynomials (i.e., such that the divided difference operator of first order applied to any polynomorphism must yield a poynomial of lower degree) can only be made on special lattices well known in contemporary $q-$calculus.
Abstract: Difference calculus compatible with polynomials (i.e., such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree) can only be made on special lattices well known in contemporary $q-$calculus. Orthogonal polynomials satisfying difference relations on such lattices are presented. In particular, lattices which are dense on intervals ($|q|=1$) are considered.
TL;DR: Bajpai as mentioned in this paper examined the validity of some orthogonality relations and expansion formulas involving a class of hypergeometric polynomials which are essentially certain modified Jacobi polynomorphisms.
1 May 1995
TL;DR: The Bernoulli polynomials Bn(x) and En(x), with n = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 28 and 28, are expressible in terms of a finite combination of trigonometric functions and the Hurwitz zeta function 4(z, a) as mentioned in this paper.
Abstract: We prove theorems on the values of the Bernoulli polynomials B,(x) with n = 2, 3, 4, ..., and the Euler polynomials En(x) with n = 1, 2, 3, ... for 0 < x < 1 where x is rational. Bn(x) and En(x) are expressible in terms of a finite combination of trigonometric functions and the Hurwitz zeta function 4(z, a). The well-known argument-addition formulae and reflection property of Bn (x) and En (x), extend this result to any rational argument. We also deduce new relations concerning the finite sums of the Hurwitz zeta function and sum some classical trigonometric series.
1 Jan 1995
TL;DR: In this paper, limit transitions between the five parameter family of Askey-Wilson polynomials, the four parameter families of big q-Jacobi polynomorphisms, and the three parameters of little qJacomorphisms in n variables associated with root system BC were derived by taking limits of q-difference operators.
Abstract: Limit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are q-analogues of the three parameter family of BC type Jacobi polynomials in n variables. The limit transitions will be derived by taking limits of q-difference operators which have these polynomials as eigenfunctions. AMS subject classification: 33D45, 33C50 Present version: November, 1996; to appear in Canad. J. Math.
TL;DR: This paper introduces a new hierarchal triangular element in which the basis functions are constructed from orthogonal polynomials—Jacobi polynmials, shown to be better conditioned than the earlier hierarchal element of Rossow and Katz.
Abstract: Hierarchal elements are finite elements which have the useful property that elements with different polynomial orders can be used together in the same mesh without causing discontinuities. This paper introduces a new hierarchal triangular element in which the basis functions are constructed from orthogonal polynomials—Jacobi polynomials. The resulting element is shown to be better conditioned than the earlier hierarchal element of Rossow and Katz.1 Recursive formulas allow the complete set of basis functions for an element to be efficiently evaluated at a given point. In addition, the formulas can be used to generate pre-computed (universal) matrices. Examples are given of universal matrices, up to order 4, for the generalized Helmholtz equation. An electromagnetic problem involving a length of transmission line is used to show the usefulness of the new elements.
TL;DR: In this paper, a generalization of the usual Sobolev-type inner product is introduced and compared with the strict diagonal case, and the zero and asymptotic properties of these kinds of polynomial sequence are studied.
TL;DR: In this article, the existence, uniqueness and characterization of polynomials orthogonal with respect to a linear, not necessarily definite, functional L defined on the set of Laurent polynomial functions is studied.