Related papers: Critical edge behavior in the singularly perturbed…
We introduce a two parameter ($\alpha, \beta>-1$) family of interacting particle systems with determinantal correlation kernels expressible in terms of Jacobi polynomials $\{ P^{(\alpha, \beta)}_k \}_{k \geq 0}$. The family includes…
The eigenvalue correlations of random matrices from the Jacobi Unitary Ensemble have a known asymptotic behavior as their size tends to infinity. In the bulk of the spectrum the behavior is described in terms of the sine kernel, and at the…
Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(\alpha_n, \beta_n)}$ is studied, assuming that $$ \lim_{n\to\infty} \frac{\alpha_n}{n}=A, \qquad \lim_{n\to\infty} \frac{\beta _n}{n}=B, $$ with…
In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight $$ w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad x\in[-1,1],\;\;\alpha>0,\;\;t>0. $$ By using the ladder…
Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad |z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for…
We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The…
We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials…
We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on $[-1,1]$. The recurrence coefficients…
This paper complements the recent investigation of \cite{DM} on the asymptotic behavior of polynomials orthogonal over the interior of an analytic Jordan curve $L$. We study the specific case of $L=\{z= w-1 +(w-1)^{-1},\ |w|=R\}$, for some…
Our goal is to find an asymptotic behavior as $n\to\infty$ of the orthogonal polynomials $P_{n}(z)$ defined by Jacobi recurrence coefficients $a_{n}$ (off-diagonal terms) and $ b_{n}$ (diagonal terms). We consider the case $a_{n}\to\infty$,…
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials p_n(z) with the quartic exponential weight exp [-N (1/2 z^2 + t/4 z^4)], where t is complex. Our goals are: A) to describe the regions of different…
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 contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are…
We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function…
We consider Jacobi matrices with eventually increasing sequences of diagonal and off-diagonal Jacobi parameters. We describe the asymptotic behavior of the subordinate solution at the top of the essential spectrum, and the asymptotic…
We prove universality at the edge of the spectrum for unitary (beta=2), orthogonal (beta=1) and symplectic (beta=4) ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial,…
Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other…
We consider the orthogonal polynomials on $[-1,1]$ with respect to the weight $$ w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \Xi_{c}(x), \quad \alpha, \beta >-1, $$ where $h$ is real analytic and strictly positive on $[-1, 1]$, and $\Xi_{c}$ is…
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials $\pi_n(z)$ with the quartic exponential weight $\exp[-N(\frac 12 z^2+\frac 14 tz^4)]$, where $t\in {\mathbb C}$ and $N\in{\mathbb N}$, $N\to\infty$. Our…
Our goal is to find an asymptotic behavior as $n\to\infty$ of orthogonal polynomials $P_{n}(z)$ defined by the Jacobi recurrence coefficients $a_{n}, b_{n}$. We suppose that the off-diagonal coefficients $a_{n}$ grow so rapidly that the…