Related papers: Commuting difference operators, spinor bundles and…
This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on…
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 obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly…
Elementary properties of the Koornwinder-Macdonald multivariable Askey-Wilson polynomials are discussed. Studied are the orthogonality, the difference equations, the recurrence relations, and the orthonormalization constants for these…
On a suitable class of non-compact manifolds, we study (pseudo)differential operators which feature an asymptotic translation-invariance along one axis and an asymptotic dilation-invariance, or asymptotic homogeneity with respect to…
Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including…
We obtain asymptotics of polynomials satisfying the orthogonality relations $$ \int_{\mathbb{R}} z^k P_n(z; t , N) \mathrm{e}^{-N \left(\frac{1}{4}z^4 + \frac{t}{2}z^2 \right)} \mathrm{d} z = 0 \quad \text{ for } \quad k = 0, 1, ..., n-1,…
We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that…
In this paper, we present a new method via the transfer matrix approach to obtain asymptotic formulae of orthogonal polynomials with asymptotically identical coefficients of bounded variation. We make use of the hyperbolicity of the…
The complex or non-hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the…
In this paper we study the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. The zeros of these polynomials are the nodes for complex Gaussian…
We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial…
Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ 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, $$ where $w(z)>0$ for…
In this paper, we are interested in matrix valued orthogonal polynomials on the real line with respect to exponential weights. We obtain strong asymptotics as the degree tends to infinity in different regions of the complex plane, as well…
We study polynomials that are orthogonal with respect to a varying quartic weight \exp(-N(x^2/2+tx^4/4)) for t<0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity,…
We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic…
We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The…
We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n,…
This paper is devoted to a study of $S$-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property ($S$-property). Such curves have many…
We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via…