Zeros Of Polynomials Research Articles

On Numerical Radius Inequalities for Hilbert Space Operators

This paper aims to find special cases of some inequalities for numerical radii and spectral radii of a bounded linear operator on a Hil-bert space, we focus on numerical radii inequalities for restricted linear operators on complex Hil-bert spaces for the case of one and two operators, and study the numerical range of an operator K on a complex Hil-bert space H, after that we present some inequalities for numerical radii and spectral radii and studied it to find new results. At the end of this paper we find several inequalities for numerical radii by using the spectral norm, this study is necessary to find other bound for zeros of polynomials and this study is necessary to find new bounds for the zeros of polynomials by a playing the new results to the companion matrix.

Interlacing of zeros of odd period polynomials

The work of Conrey, Farmer and Imamoglu shows that all nontrivial zeros of the odd period polynomial of a Hecke eigenform of level one lie on the unit circle. We further show that nontrivial zeros of odd period polynomials of Hecke eigenforms of different weights share certain interlacing property.

A degenerate version of hypergeometric Bernoulli polynomials: announcement of results

Abstract This article explores some properties of degenerate hypergeometric Bernoulli polynomials, which are defined through the following generating function t m e λ x ( t ) e λ x ( t ) - ∑ l = 0 m - 1 ( 1 ) l , λ t l l ! = ∑ n = 0 ∞ B n , λ [ m - 1 ] ( x ) t n n ! , | t | < min { 2 π , 1 | λ | } , λ ∈ ℝ \ { 0 } . {{{t^m}e_\lambda ^x\left( t \right)} \over {e_\lambda ^x\left( t \right) - \sum\nolimits_{l = 0}^{m - 1} {\left( 1 \right)l,\lambda{{{t^l}} \over {l!}}} }} = \sum\limits_{n = 0}^{^\infty } {B_{n,\lambda }^{\left[ {m - 1} \right]}} \left( x \right){{{t^n}} \over {n!}},\,\,\,\,\left| t \right| < \min \left\{ {2\pi ,{1 \over {\left| \lambda \right|}}} \right\},\lambda \in \mathbb{R}\backslash \left\{ 0 \right\}. We deduce their associated summation formulas and their corresponding determinant form. Also we focus our attention on the zero distribution of such polynomials and perform some numerical illustrative examples, which allow us to compare the behavior of the zeros of degenerate hypergeometric Bernoulli polynomials with the zeros of their hypergeometric counterpart. Finally, using a monomiality principle approach we present a differential equation satisfied by these polynomials.

On the structure of the matrix in Sturm algorithm for determining the number of zeros in the open unit disk for a real polynomial

In 1984, C.W Schelin presented a numerical method for calculating the number of zeros of a real polynomial inside the unit disk using the argument principle and Cauchy index on [-1, 1] inspired by sturm’s algorithm [3], this method is based on calculation of the generalized Sturm sequence in Chebyshev form, however in some cases this algorithm does not efficient for this reason, it has been modified in another works [1], [2]. There are other numerical algorithms for calculating the number of zeros of a polynomial in the unit disk like shur -Cohn and Marden-Cohn but the numerical method presented in this study [1] is more efficient in cost of elementary operations (o (n2)). In this paper, we propose to improve this algorithm [1], to do this we study the matrix structure involved in the calculation of chebychev polynomials, for there on we prove that is a Toeplitz matrix which is known for their important properties in the complexity of calculations and we calculate its last line which is necessary to define the whole matrix to solve the linear system for calculating polynomials which are necessary to obtain Chebyshev sequence, this result will reduce calculation steps in this method [1]. This numerical algorithm may operate to study the stability of dynamical systems [2].

Concyclicity of the zeros of polynomials associated to derivatives of the L-functions of Eisenstein series

Concyclicity of the zeros of polynomials associated to derivatives of the L-functions of Eisenstein series

On the differential equations of frozen Calogero-Moser-Sutherland particle models

Multivariate Bessel and Jacobi processes describe Calogero-Moser-Sutherland particle models. They depend on a parameter k and are related to time-dependent classical random matrix models like Dysom Brownian motions, where k has the interpretation of an inverse temperature. There are several stochastic limit theorems for k→∞ were the limits depend on the solutions of associated ordinary differential equations (ODEs) where these ODEs admit particular simple solutions which are connected with the zeros of the classical orthogonal polynomials. In this paper we show that these solutions attract all solutions. Moreover we present a connection between the solutions of these multivariate ODEs with associated one-dimensional inverse heat equations. These inverse heat equations are used to compute the expectations of some determinantal formulas for the multivariate Bessel and Jacobi processes.

Generalized Gauss-Rys orthogonal polynomials

Let (Pn(x;z;λ))n≥0 be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional s defined by〈s,p〉=∫−11p(x)(1−x2)(λ−1/2)e−zx2dx,λ>−1/2,z>0. In this contribution, several properties of the polynomials Pn(x;z;λ) are studied taking into account the relation between the parameters of the three-term recurrence relation that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear naturally. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameters z and λ are given.

Steady Boussinesq convection: Parametric analyticity and computation

AbstractSteady solutions to the Navier–Stokes equations with internal temperature forcing are considered. The equations are solved in two dimensions using the Boussinesq approximation to couple temperature and density fluctuations. A perturbative Stokes expansion is used to prove that that steady flow variables are parametrically analytic in the size of the forcing. The Stokes expansion is complemented with analytic continuation, via functional Padé approximation. The zeros of the denominator polynomials in the Padé approximants are observed to agree with a numerical prediction for the location of singularities of the steady flow solutions. The Padé representations not only prove to be good approximations to the true flow solutions for moderate intensity forcing, but are also used to initialize a Newton solver to compute large amplitude solutions. The composite procedure is used to compute steady flow solutions with forcing several orders of magnitude larger than the fixed‐point method developed in previous work.

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

AbstractIn this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.

Values of p-adic hypergeometric functions and p-adic analogue of Kummer’s linear identity

Let p be an odd prime and [Formula: see text] be the finite field with p elements. This paper focuses on the study of the values of a generic family of hypergeometric functions in the p-adic setting, which we denote by [Formula: see text], where [Formula: see text] and [Formula: see text]. These values are expressed in terms of numbers of zeros of certain polynomials over [Formula: see text]. These results lead to certain p-adic analogues of classical hypergeometric identities. Namely, we obtain p-adic analogues of particular cases of a Gauss’s theorem and a Kummer’s theorem. Moreover, we examine the zeros of these functions. For example, if n is odd, we characterize t for which [Formula: see text] has zeros. In contrast, we show that if n is even, then the function [Formula: see text] has no zeros for any prime p apart from the trivial case when [Formula: see text].

On the zeros of polyanalytic polynomials

We give sufficient conditions under which a polyanalytic polynomial of degree n has (i) at least one zero, and (ii) finitely many zeros. In the latter case, we prove that the number of zeros is bounded by n2. We then show that for all k∈{0,1,2,…,n2,∞} there exists a polyanalytic polynomial of degree n with exactly k distinct zeros. Moreover, we generalize the Lagrange and Cauchy bounds from analytic to polyanalytic polynomials and obtain inclusion disks for the zeros. Finally, we construct a harmonic and thus polyanalytic polynomial of degree n with n nonzero coefficients and the maximum number of n2 zeros.

Riemann hypothesis for period polynomials for cusp forms on Γ0(N)

We prove that for even integer k, almost all of zeros of the period polynomial associated to a cusp form of weight k on [Formula: see text] are on the circle [Formula: see text] under some conditions.

An extension of a mixed interpolation–regression method using zeros of orthogonal polynomials

The constrained mock-Chebyshev least squares approximation (CMCLS-approximation) is a method that has been recently introduced. It operates on a grid of equidistant points, aiming to eliminate the Runge phenomenon. The implementation of the idea behind this approximation method involves interpolating the function exclusively on the subset of nodes closer to the set of Chebyshev–Lobatto nodes of a suitable order and using the remaining nodes to enhance the accuracy of the approximation through a simultaneous regression. The main goal of this article is to extend the CMCLS-approximation through the interpolation on zeros of orthogonal polynomials, leveraging their inherent favorable properties.

Upper bounds for the spectral radius of matrices having the Perron–Frobenius property

A new upper bound for the spectral radius of matrices having the Perron–Frobenius property is given by considering the position of positive entries. Some examples involving the largest zero of polynomials and the spectral radius of the iterative matrix for the Perron–Frobenius splitting are given to show the superiority of the theoretical result.

Sturm’s Comparison Theorem for Classical Discrete Orthogonal Polynomials

In an earlier work (Castillo et al. in J Math Phys 61:103505, 2020), it was established, from a hypergeometric-type difference equation, tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In this work, we continue with the study of zeros of these polynomials by giving a comparison theorem of Sturm type. As an application, we analyze in a simple way some relations between the zeros of certain classical discrete orthogonal polynomials.

Probabilistic zero bounds of certain random polynomials

Abstract This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy’s bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials.

When D-companion matrix meets incomplete polynomials

In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we obtain a majorization relationship between the zeros of convex combinations of incomplete polynomials and the original polynomial. Moreover, we prove that the set of zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. Location of zeros of convex combinations of incomplete polynomials is determined.

Effect of the zeros of initial term non-gap polynomials of higher index on the uniqueness of meromorphic functions

Effect of the zeros of initial term non-gap polynomials of higher index on the uniqueness of meromorphic functions

Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Construction and Convergence of H-S Combined Mean Method for Multiple Polynomial Zeros

Expected Energy of Zeros of Elliptic Polynomials

AbstractIn 2011, Armentano, Beltrán and Shub obtained a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different approach which allows us to extend the study to the Riesz energies and to compute the expected separation distance.