Author
Luc Vinet
Other affiliations: McGill University, Université de Montréal, University of California, Los Angeles ...read more
Bio: Luc Vinet is an academic researcher from Centre de Recherches Mathématiques. The author has contributed to research in topics: Orthogonal polynomials & Classical orthogonal polynomials. The author has an hindex of 43, co-authored 438 publications receiving 8025 citations. Previous affiliations of Luc Vinet include McGill University & Université de Montréal.
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TL;DR: In this article, a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type was studied.
Abstract: We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi polynomials in the limit $q=-1$. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for $q=-1$.
775 citations
TL;DR: In this article, a family of orthogonal polynomials which satisfy (apart from a three-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl type was studied.
Abstract: We study a family of 'classical' orthogonal polynomials which satisfy (apart from a three-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl type. These polynomials can be obtained from the little q-Jacobi polynomials in the limit q = −1. We also show that these polynomials provide a nontrivial realization of the Askey–Wilson algebra for q = −1.
564 citations
TL;DR: In this article, a formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived.
Abstract: The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.
274 citations
01 Jan 2000
TL;DR: In this article, the authors present a generalization of the Calogero-Moser-Sutherland model for the C-Invariant Tensors R-Matrices.
Abstract: Classical Dynamics r-Matrices for Calogero-Moser Systems and Their Generalizations Hidden Algebraic Structure of Calogero-Sutherland Model Polynomial Eigenfunctions of the Calogero-Sutherland-Moser Models The Theory of Lacunas and Quantum Integrable Systems Canonical Forms for the C-Invariant Tensors R-Matrices, Generalized Inverses and Calogero-Moser-Sutherland Models Tricks of the Trade:. Classical and Quantum Partition Functions of the Calogero-Moser-Sutherland Model The Meander Determinant and its Generalizations Differential Equations for Multivariable Hermite and Laguerre Polynomials Quantum Currents Realizaton of the Elliptic Quantum Groups Heisenberg-Ising Spin Chain:.. - Ruijsenaars' Commuting Difference System from Belavin's Elliptic R-Matrix Invariants and Eigenvectors for quantum Heisenberg Chains with Elliptic Exchanges The Bispectral Involution as a Linearizing Map On Some Quadratic Algebras:. Elliptic Solutions to Difference Nonlinear Equations and Nested Bethe Ansatz Equations Creation.
157 citations
TL;DR: In this paper, a study of non-relativistic superintegrable systems whose invariants are quadratic in the momenta is presented, and the symmetries responsible for the accidental degeneracies of those problems are investigated and described in terms of polynomial algebras.
140 citations
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Journal Article•
28,685 citations
TL;DR: In this article, the authors review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications, including shape invariance and operator transformations, and show that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials.
2,688 citations
Posted Content•
TL;DR: The Askey-scheme of hypergeometric orthogonal polynomials was introduced in this paper, where the q-analogues of the polynomial classes in the Askey scheme are given.
Abstract: We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme.
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.
1,459 citations
TL;DR: In this article, the authors studied a random Groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process and showed that shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
1,031 citations