Related papers: Zeros of Multiple Orthogonal Polynomials: Location…
A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval $(0,1)$ is studied. This type of polynomials have direct applications in…
In this work, the explicit expressions of coefficients involved in quasi Christoffel polynomials of order one and quasi-Geronimus polynomials of order one are determined for Jacobi polynomials. These coefficients are responsible for…
We study the asymptotic zero distribution of type II multiple orthogonal polynomials associated with two Macdonald functions (modified Bessel functions of the second kind). Based on the four-term recurrence relation, it is shown that, after…
Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi_0,\xi_1,\ldots,\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its…
We state and prove the $q$-extension of a result due to Johnston and Jordaan (cf. \cite{Johnston-2015}) and make use of this result, the orthogonality of $q$-Laguerre, little $q$-Jacobi, $q$-Meixner and Al-Salam-Carlitz I polynomials as…
We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…
The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix…
For a given polynomial $P$ with simple zeros, and a given semiclassical weight $w$, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of $P$. The…
The complex or non-hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the…
We study multiple orthogonal polynomials of type I and type II which have orthogonality conditions with respect to r measures. These polynomials are connected by their recurrence relation of order r+1. First we show a relation with the…
We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq k, 0\leq k…
In this note, we provide a wide range of upper bounds for the moduli of the zeros of a complex polynomial. The obtained bounds complete a series of previous papers on the location of zeros of polynomials.
Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large $n$. Motivated by two plots, we look at the finer structure for the case of random Verblunsky coefficients and for what…
Multiple orthogonal polynomials are traditionally studied because of their connections to number theory and approximation theory. In recent years they were found to be connected to certain models in random matrix theory. In this paper we…
We prove some new results about the spacing between neighboring zeros of paraorthogonal polynomials on the unit circle. Our methods also provide new proofs of some existing results. The main tool we will use is a formula for the phase of…
We investigate type I multiple orthogonal polynomials on $r$ intervals which have a common point at the origin and endpoints at the $r$ roots of unity $\omega^j$, $j=0,1,\ldots,r-1$, with $\omega = \exp(2\pi i/r)$. We use the weight…
In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for…
We study a family of type II multiple orthogonal polynomials. We consider orthogonality conditions with respect to a vector measure, in which each component is a q-analogue of the binomial distribution. The lowering and raising operators as…
We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these…
Given an orthogonal polynomial sequence on the real line, another sequence of polynomials can be found by composing these polynomials with a general M\"obius transformation. In this work, we study the properties of such M\"obius-transformed…