Related papers: Asymptotics of multivariate orthogonal polynomials…
We study the Fourier transform of polynomials in an orthogonal family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier--Bessel functions, in the argument, the…
The linearization coefficients for a set of orthogonal polynomials are given explicitly as a weighted sum of combinatorial objects. Positivity theorems of Askey and Szwarc are corollaries of these expansions.
We discuss the relationship between ratio asymptotics for general orthogonal polynomials and the asymptotics of the associated Bergman shift operator. More specifically, we consider the case in which a measure is supported on an infinite…
We establish the Plancherel-Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases,…
Given a bivariate weight function defined on the positive quadrant of $\mathbb{R}^2$, we study polynomials in two variables orthogonal with respect to varying measures obtained by special modifications of this weight function. In…
We consider the Plancherel-Rotach type asymptotics of the biorthogonal polynomials associated to the biorthogonal ensemble with the joint probability density function \begin{equation*} \frac{1}{C} \prod_{1 \leq i < j \leq n} (\lambda_j…
We derive Mehler--Heine type asymptotic formulas for Charlier and Meixner polynomials, and also for their associated families. These formulas provide good approximations for the polynomials in the neighborhood of $x=0,$ and determine the…
In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlev\'e transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the…
We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and…
We address the problem of the weak asymptotic behavior of zeros of families of generalized hypergeometric polynomials as their degree tends to infinity. The main tool is the representation of such polynomials as a finite free convolution of…
Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and…
In this paper we investigate algebraic, differential and asymptotic properties of polynomials $p_n(x)$ that are orthogonal with respect to the complex oscillatory weight $w(x)=e^{i\omega x}$ on the interval $[-1,1]$, where $\omega>0$. We…
The classical orthogonal polynomials (Hermite, Laguerre and Jacobi) are involved in a vast number of applications in physics and engineering. When large degrees $n$ are needed, the use of recursion to compute the polynomials is not a good…
The plane partition polynomial $Q_n(x)$ is the polynomial of degree $n$ whose coefficients count the number of plane partitions of $n$ indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these…
In this paper cycles for asymptotic solutions for Heckman-Opdam hypergeometric system of type $A_n$ are described. Cycles are enumerated by elements of symmetric group. Leading asymptotic and leading coefficient are calculated. Value of…
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence…
In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre…
Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large…
We present an asymptotic analysis of the Verblunsky coefficients for the polynomials orthogonal on the unit circle with the varying weight $e^{-nV(\cos x)}$, assuming that the potential $V$ has four bounded derivatives on $[-1,1]$ and the…
We obtain large n asymptotics for products of powers of the absolute values of the characteristic polynomials in the Gaussian Unitary Ensemble of n\times n matrices. Our results can also be interpreted as asymptotics of the determinant of a…