Limits for circular Jacobi beta-ensembles

Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the circular Jacobi $\beta$-ensemble, which is a generalization of the Dyson circular $\beta$-ensemble but equipped with an additional parameter $b$, and further studied its limiting spectral measure. We calculate the scaling limits for expected products of characteristic polynomials of circular Jacobi $\beta$-ensembles. For the fixed constant $b$, the resulting limit near the spectrum singularity is proven to be a new multivariate function. When $b=\beta Nd/2$, the scaling limits in the bulk and at the soft edge agree with those of the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\beta$-ensembles proved in the joint work with P Desrosiers "Asymptotics for products of characteristic polynomials in classical beta-ensembles", Constr. Approx. 39 (2014), arXiv:1112.1119v3. As corollaries, for even $\beta$ the scaling limits of point correlation functions for the ensemble are given. Besides, a transition from the spectrum singularity to the soft edge limit is observed as $b$ goes to infinity. The positivity of two special multivariate hypergeometric functions, which appear as one factor of the joint eigenvalue densities for spiked Jacobi/Wishart $\beta$-ensembles and Gaussian $\beta$-ensembles with source, will also be shown.Comment: 26 page

Holomorphic Hermite polynomials in two variables

Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in $z$ and $\bar{z}$ which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. "Homotopic" behavior of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in $z$ and $\bar{z}$ on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in $z$ and $\bar{z}$

Orthogonal Polynomials and Sharp Estimates for the Schr\"odinger Equation

In this paper we study sharp estimates for the Schr\"odinger operator via the framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer polynomials to prove a new weighted inequality for the Schr\"odinger equation that is maximized by radial functions. We use Hermite and Laguerre polynomial expansions to produce sharp Strichartz estimates for even exponents. In particular, for radial initial data in dimension 2, we establish an interesting connection of the Strichartz norm with a combinatorial problem about words with four letters.Comment: 22 page

Classes of Bivariate Orthogonal Polynomials

We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-$D$ Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give $q$-analogues of all these extensions. In each case in addition to generating functions and three term recursions we provide raising and lowering operators and show that the polynomials are eigenfunctions of second-order partial differential or $q$-difference operators

The Matrix Ansatz, Orthogonal Polynomials, and Permutations

In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We illustrate this approach with applications to moments of orthogonal polynomials, permutations, signed permutations, and tableaux.Comment: to appear in Advances in Applied Mathematics, special issue for Dennis Stanto

Orthogonality of Hermite polynomials in superspace and Mehler type formulae

In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry G < O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m) x Sp(2n) are considered. After an overview of the results for O(m) and G, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proven. Next, Hermite polynomials in a full superspace with O(m) x Sp(2n) symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schroedinger operators in superspace. Subsequently, a Mehler formula for the full superspace is obtained, thus yielding an eigenfunction decomposition of the super Fourier transform. Finally, an extensive comparison is made of the results in the different types of symmetry.Comment: Proc. London Math. Soc. (2011) (42pp

The Kontorovich-Lebedev transform as a map between dd-orthogonal polynomials

A slight modification of the Kontorovich-Lebedev transform is an automorphism on the vector space of polynomials. The action of this $KL_{\alpha}$-transform over certain polynomial sequences will be under discussion, and a special attention will be given the d-orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the $KL_{\alpha}$-transform of a 2-orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose $KL_{\alpha}$-transform is a $d$-orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and $d$ is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.Comment: 27 page