Related papers: On skyburst polynomials and their zeros
The zeros of semi-orthogonal functions with respect to a probability measure mu supported on the unit circle can be applied to obtain Szego quadrature formulas. The discrete measures generated by these formulas weakly converge to the…
Consider random polynomials of the form $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d.\ non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$…
In recent years, chain sequences and their perturbations have played a significant role in characterising the orthogonal polynomials both on the real line as well as on the unit circle. In this note, a particular disturbance of the chain…
In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and…
A class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials do in the theory of Bessel functions. The measure of orthogonality for this new…
We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal…
Asymptotic behavior of orthogonal polynomials on the circle, with respect to a weight having a fractional zero on the torus. Applications to the eigenvalues of certain unitary random matrices. This paper is devoted to the orthogonal…
For any finite partially ordered set $P$, the $P$-Eulerian polynomial is the generating function for the descent number over the set of linear extensions of $P$, and is closely related to the order polynomial of $P$ arising in the theory of…
We investigate the type I and type II multiple orthogonal polynomials on an $r$-star with weight function $|x|^{\beta}e^{-x^r}$, with $\beta>-1$. Each measure $\mu_j$, for $1\leq j \leq r$, is supported on the semi-infinite interval…
A classical result of Erdos and Turan states that if a monic polynomial has small size on the unit circle and its constant coefficient is not too small, then its zeros cluster near the unit circle and become equidistributed in angle. Using…
We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic…
We study the limiting zero distribution of orthogonal polynomials with respect to some particular exponential weights exp(-nV(z)) along contours in the complex plane. We are especially interested in the question under which circumstances…
We survey results on the distribution of zeros of random polynomials and of random holomorphic sections of line bundles, especially for large classes of probability measures on the spaces of holomorphic sections. We provide furthermore some…
We study orthogonal polynomials on a fully symmetric planar domain $\Omega$ that is generated by a certain triangle in the first quadrant. For a family of weight functions on $\Omega$, we show that orthogonal polynomials that are even in…
We consider polynomials $p_n^{\omega}(x)$ that are orthogonal with respect to the oscillatory weight $w(x)=e^{i\omega x}$ on $[-1,1]$, where $\omega>0$ is a real parameter. A first analysis of $p_n^{\omega}(x)$ for large values of $\omega$…
In this paper, we obtain several new classes of irreducible polynomials having integer coefficients whose zeros lie inside an open disk around the origin or outside a closed annular region in the complex plane. Such irreducible polynomials…
A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical…
From a sequence $\left\{ a_{n}\right\} _{n=0}^{\infty}$ of real numbers satisfying a three-term recurrence, we form a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ whose coefficients are numbers in this sequence. We…
Explicit examples of {\bf positive} crystalline measures and Fourier quasicrystals are constructed using pairs of stable of polynomials, answering several open questions in the area.
We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$…