相关论文: Non-Symmetric Jack Polynomials and Integral Kernel…
We investigate the existence and the orthogonality of the generalized Jack symmetric functions which play an important role in the AGT relations. We show their orthogonality by deforming them to the generalized Macdonald symmetric…
We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of…
We construct a family of pairwise commuting operators such that the Jack symmetric functions of infinitely many variables $x_1,x_2,...$ are their eigenfunctions. These operators are defined as limits at $N\to\infty$ of renormalised…
We consider the asymptotic local behavior of the second correlation function of the characteristic polynomials of sparse non-Hermitian random matrices $X_n$ whose entries have the form $x_{jk}=d_{jk}w_{jk}$ with iid complex standard…
In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial…
The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a…
We consider a realization of fractional supersymmetric of quantum mechanics of order $r$, where the Hamiltonian and supercharges involve reflection operators. It is shown that the Hamiltonian has $r$-fold degenerate spectrum and the…
Two families (type $A$ and type $B$) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri type recurrence formulas for these families. In the…
We study the coinvariant ring of the complex reflection group $G(r,p,n)$ as a module for the corresponding rational Cherednik algebra $\HH$ and its generalized graded affine Hecke subalgebra $\mathcal{H}$. We construct a basis consisting of…
We discuss as a fundamental characteristic of orthogonal polynomials like the existence of a Lie algebra behind them, can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we put…
The plethystic transformation $f[X] \mapsto f[X/(1-t)]$ and LLT polynomials are central to the theory of symmetric Macdonald polynomials. In this work, we introduce and study nonsymmetric flagged LLT polynomials. We show that these admit…
We use generalized kernel functions to construct explicit solutions by integrals of the non-stationary Schr\"odinger equation for the Hamiltonian of the elliptic Calogero-Sutherland model (also known as elliptic…
Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they…
Within the context of Supersymmetric Quantum Mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the…
Decay rates for the sequence of eigenvalues of positive and compact integral operators has been largely investigated for a long time in the literature. In this paper, the focus will be on positive integral operators acting on square…
We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal…
Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is…
We review some recents developments of the algebraic structures and spectral properties of non-Hermitian deformations of Calogero models. The behavior of such extensions is illustrated by the $A_2$ trigonometric and the $D_3$ angular…
The Hermite functions are an orthonormalbasis of the space of square integrable functions with favourable approximation properties. Allowing for a flexible localization in position and momentum, the Hagedorn wavepackets generalize the…
In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted as $\{\mathbb{H}_{n}(x;q)\}_{n\geq 0}$, which are orthogonal with respect to the following non-standard inner…