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Preliminary version of Chapter 2 in the book "Encyclopedia of Special functions: The Askey-Bateman Project, Vol. 2: Multivariate special functions", T. H. Koornwinder and J. V. Stokman (eds.), Cambridge University Press, 2021.
We study the stability of the mesoscopic fluctuations of certain orthogonal polynomial ensembles on the real line utilizing the recurrence relation of the associated orthogonal polynomials. We prove that under a sparse enough decaying…
Interacting Fock space connects the study of quantum probability theory, classical random variables, and orthogonal polynomials. It is a pre-Hilbert space associated with creation, preservation, and annihilation processes. We prove that if…
We use a non-linear characterization of orthonormal polynomials due to Saff in order to show that the behavior of orthonormal polynomials is determined only by its leading coefficient and its normalization. Several applications of this…
Recently, we adapted random walk arguments based on work of Nachmias and Peres, Martin-L\"of, Karp and Aldous to give a simple proof of the asymptotic normality of the size of the giant component in the random graph $G(n,p)$ above the phase…
The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few…
There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [11]. We show how those results can be understood using ideas from…
In this paper, we study the asymptotic behavior of a semi-linear slow-fast stochastic partial differential equation with singular coefficients. Using the Poisson equation in Hilbert space, we first establish the strong convergence in the…
We study normality of a family of meromorphic functions, whose differential polynomials satisfy a certain condition, which significantly improves and generalizes some recent results of Chen (Filomat, 31(14) 2017, 4665-4671). Moreover, we…
This paper is dedicated to the investigation of the asymptotic behavior of a delayed wave equation without the presence of any position term. First, it is shown that the problem is well-posed in the sense of semigroups theory. Thereafter,…
We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients $\alpha_{n,N}$ when the ratio $n/N$ converges as $n,N\to\infty$. First, we give a streamlined proof of ratio asymptotics for…
We obtain explicit upper and lower bounds on the norms of the spectral projections of the non-self-adjoint harmonic oscillator. Some of our results apply to a variety of other families of orthogonal polynomials.
We give the asymptotic behavior of the ratio of two neighboring multiple orthogonal polynomials under the condition that the recurrence coefficients in the nearest neighbor recurrence relations converge.
It is our aim in this note to give a counter example to an argument used in the proof of the main theorem of the paper: On iterations for families of asymptotically pseudocontractive mappings, Applied Mathematics Letters, 24 (2011), 33-38…
Based on the work of Chen and Its [{\em J. Approx. Theory} {\bf 162} ({2010}) {270--297}], we further study orthogonal polynomials with respect to the singularly perturbed Laguerre weight $w(x;t,\alpha) = {x^\alpha}{\mathrm e^{-…
In this paper, we present a new method via the transfer matrix approach to obtain asymptotic formulae of orthogonal polynomials with asymptotically identical coefficients of bounded variation. We make use of the hyperbolicity of the…
In the first five sections, we deal with the class of probability measures with asymptotically periodic Verblunsky coefficients of p-type bounded variation. The goal is to investigate the perturbation of the Verblunsky coefficients when we…
We consider polynomials orthogonal on $[0,\infty)$ with respect to Laguerre-type weights $w(x)=x^\alpha e^{-Q(x)}$, where $\alpha>-1$ and where $Q$ denotes a polynomial with positive leading coefficient. The main purpose of this paper is to…
Asymptotic representation of minimal polynomials on several intervals is given. The last modifications and corrections of this manuscript were done by the author in the two months preceding his passing away in November 2009. The manuscript…
This is the full version of "Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results" appearing on this server and also…