Zeros Of Classical Orthogonal Polynomials Research Articles

Bounds for Extreme Zeros of Classical Orthogonal Polynomials Related to Birth and Death Processes

In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related chain sequences. Using these chain sequences, we find the transition probabilities for the corresponding process. As a consequence, transition probabilities related to G-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results.

Stieltjes’ theorem for classical discrete orthogonal polynomials

The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, quantum q-Krawtchouk, q-Krawtchouk, affine q-Krawtchouk, q-Charlier, Al-Salam–Carlitz, q-Hahn, little q-Jacobi, little q-Laguerre/Wall, q-Bessel, q-Racah, and dual q-Hahn polynomials.

Perturbations around the zeros of classical orthogonal polynomials

Starting from degree N solutions of a time dependent Schrödinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree (0,1,…,N−1) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.

Zeros of classical orthogonal polynomials of a discrete variable

In this paper we obtain sharp bounds for the zeros of classical orthogonal polynomials of a discrete variable, considered as functions of a parameter, by using a theorem of A. Markov and the so-called HellmannFeynman theorem. Comparisons with previous results for zeros of Hahn, Meixner, Kravchuk and Charlier polynomials are also presented.

Sharp bounds for the extreme zeros of classical orthogonal polynomials

Bounds for the extreme zeros of the classical orthogonal polynomials are obtained by a surprisingly simple method. Nevertheless, it turns out that, in most cases, the estimates obtained in this note are better than the best limits known in the literature.

On extreme zeros of classical orthogonal polynomials

Let x 1 and x k be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree k . We shall establish sharp inequalities of the form x 1 < A , x k > B , which are uniform in all the parameters involved. Together with inequalities in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with a high precision.

Location of the zeros of polynomials satisfying three-term recurrence relations with complex coefficients

Two papers on the same subject were pubkushed by the authors in 1985:part I[1]and part III[2].The method announced for part II did not give satisfying results.However,we noticed that our method of estimationg the boundaries of zero-free regions for polynomials can be improved by introducing proper optimizing paramcters for each polynomial.The optimization is then performed over all these parameters at the end. It was showed that this finest method of optimimation is sharp in the following sense:it leads to the exact interval where all zeros of classical orthogonal polynomials are licated[3].In the present paper we give the complete proofs of these improved theorems of lication in quaestion.

Numerical Integration--Recent Developments, Software and Applications.

Theoretical Aspects of One-Dimensional Quadrature.- The convergence of noninterpolatory product integration rules.- Some quadrature rules for finite trigonometric and related integrals.- Modified Clenshaw-Curtis integration and applications to numerical computation of integral transforms.- Fast generation of quadrature rules with some special properties.- Quadrature methods for the determination of zeros of transcendental functions - a review.- Computation of the index of an analytic function.- Approximate computation of strongly oscillatory integrals with compound precision.- Bounds and approximations for the zeros of classical orthogonal polynomials. Their use in generating quadrature rules..- Indefinite integration formulas based on the sinc expansion.- On positive quadrature rules.- Quadrature rules with end-point corrections - comments on a paper by Garloff, Solak and Szydelko.- Comparison of Gauss-Hermite and midpoint quadrature with application to the Voigt function.- Theoretical Aspects of Multiple Quadrature.- On sequences of imbedded integration rules.- Asymptotic expansions and their applications in numerical integration.- Construction of three-dimensional cubature formulae with points on regular and semi-regular polytopes.- Construction of sequences of embedded cubature formulae for circular symmetric planar regions.- On the construction of higher degree three dimensional embedded integration rules.- Fully symmetric integration rules for the unit four-cube.- On the construction of cubature formulae with few nodes using Groebner bases.- Quasi-random sequences for optimization and numerical integration.- Non-fortuitous, non-product, non-fully symmetric cubature structures.- Algorithms, Software and Applications.- A parallelization of adaptive integration methods.- The numerical evaluation of multiple integrals on parallel computers.- Vectorisation of one dimensional quadrature codes.- Performance of self-validating adaptive quadrature.- Using weight functions in self-validating quadrature.- On the construction of a practical Ermakov-Zolotukhin multiple integrator.- Interactive numerical quadrature.- The design of a generic quadrature library in Ada.- The integration of the multivariate normal density function for the triangular method.- Numerical integration in scalar wave scattering, with application to acoustic scattering by fish.- Software Classification and Testing.- Testing multiple integrators.- A package for testing multiple integration subroutines.- A test of some well known one dimensional general purpose automatic quadrature routines.- Development of useful quadrature software, with particular emphasis on microcomputers.- Appendix I: Final Program.- Appendix II: Discussion.- List of Participants.- List of Contributors.

A general technique to obtain nonlinear equations for the zeros of classical orthogonal polynomials

A general technique to obtain nonlinear equations for the zeros of classical orthogonal polynomials