On Eisenstein polynomials and zeta polynomials

Eisenstein polynomials, which were defined by Oura, are analogues of the concept of an Eisenstein series. Oura conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In this paper, we provide new analogous properties of Eisenstein polynomials and zeta polynomials. These properties are finite analogies of certain properties of Eisenstein series.     

On connection coefficients of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

Orthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation.     

涉及CM分担值的常系数指数多项式的唯一性

主要证明了两个非常数的常系数指数多项式,如果在复平面中4个张角严格大于π的每一个角域内都有1个有穷的CM分担值,且这4个分担值是判别的,则它们必然恒等.     

Elementary symmetric polynomials in Shamir's scheme

The concept of k-admissible tracks in Shamir's secret sharing scheme over a finite field was introduced by Schinzel et al. (2009) [10]. Using some estimates for the elementary symmetric polynomials, we show that the track (1,…,n) over Fp is practically always k-admissible; i.e., the scheme allows to place the secret as an arbitrary coefficient of its generic polynomial even for relatively small p. Here k is the threshold and n the number of shareholders.     

On a class of equilibrium problems in the real axis

In a series of seminal papers, Thomas J. Stieltjes (1856-1894) gave an elegant electrostatic interpretation for the zeros of classical families of orthogonal polynomials, such as Jacobi, Hermite and Laguerre polynomials. More generally, he extended this approach to the zeros of polynomial solutions of certain second-order linear differential equations (Lamé equations), the so-called Heine-Stieltjes polynomials.In this paper, a class of electrostatic equilibrium problems in R, where the free unit charges x₁,…,x_n∈R are in presence of a finite family of “attractors” (i.e., negative charges) z₁,…,z_m∈C?R, is considered and its connection with certain class of Lamé-type equations is shown. In addition, we study the situation when both n→∞ and m→∞, by analyzing the corresponding (continuous) equilibrium problem in presence of a certain class of external fields.     

Uniqueness of Gaussian quadrature formula for computed tomography

This note answers positively a question raised by B. Bojanov and G. Petrova. Namely, the Gaussian quadrature formula for computed tomography among the given type is unique.     

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.