the recurrence relations fullled by systems of orthogonal polynomials on
the unit circle. In particular, we discuss the question, if the relevant links are con-
sistent with the recurrence relations or not. This leads to some new insight into
the analyzed classes of polynomials.
Keywords: Orthogonal polynomials on the unit circle, recurrence relations of
Szegő-type, theorem of Eneström–Kakeya
MSC 2010: Primary 42C05; secondary 30C15
Andreas Lasarow:Mathematisches Institut, Universität Leipzig, Postfach 10 09 20,
04009 Leipzig, Germany, e-mail: lasarow
Given a real function f on an interval [a, b] satisfying mild regularity conditions, we determine the number of zeros of f by evaluating a certain integral. The integrand depends on f, f′ and f″. In particular, by approximating the integral with the trapezoidal rule on a fine enough grid, we can compute the number of zeros of f by evaluating finitely many values of f, f′ and f″. A variant of the integral even allows to determine the number of the zeros broken down by their multiplicity.
The classical Eneström-Kakeya theorem establishes explicit upper and lower bounds on the zeros of a polynomial with positive coefficients and has been generalized for positive definite matrix polynomials by several authors. Recently, extensions that improve the (scalar) Eneström-Kakeya theorem were obtained with a transparent and unified approach using just a few tools. Here, the same tools are used to generalize these extensions to positive definite matrix polynomials, while at the same time generalizing the tools themselves. In the process, a framework is developed that can naturally generate additional similar results.
Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.
Vieta’s classical formulae explicitly determine the coefficients of a polynomial p ∈ 𝔽[x] in terms of the roots of p, where 𝔽 is any commutative ring. In this paper, Vieta’s formulae are obtained for slice-regular polynomials over the noncommutative algebra of quaternions, by an argument which essentially relies on induction, without invoking quasideterminants or noncommutative symmetric functions.
We prove that sausages are the family of ‘extremal sets’ in relation to certain linear improvements of Minkowski’s first inequality when working with projection/sections assumptions. In particular they characterize the equality cases of the corresponding linear refinements of both the isoperimetric inequality and Urysohn’s inequality. We also characterize sausages by algebraic properties of the roots of Steiner polynomials, in which other functionals of convex bodies such as the inradius, the mean width or the diameter are involved.
In this investigation, our main objective is to ascertain the radii of k-uniform convexity of order and the radii of strong starlikeness of the some normalized q-Bessel and Wright functions. In making this investigation we deal with the normalized Wright functions for three different kinds of normalization and six different normalized forms of q-Bessel functions. The key tools in the proof of our main results are the Mittag-Leffer expansion for Wright and q-Bessel functions and properties of real zeros of these functions and their derivatives. We also have shown that the obtained radii are the smallest positive roots of some functional equations.
The survey collects many recent advances on area Nevanlinna type classes and related spaces of analytic functions in the unit disk concern- ing zero sets and factorization representations of these classes and discusses approaches, used in proofs of these results.
Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers tends to 1. We prove that the Conjecture of Lehmer is true for by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn, zj,n, of Gn(X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for , with a minoration of the house , and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots of Gn, near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.