arXiv subjects

Luc Vinet

Publications and source records attributed to Luc Vinet.

At least 181 records · Page 10Linked to original sources

arXiv2015

-1 Krall-Jacobi Polynomials

We study a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these polynomials consists in the continuous measure of the little -1 Jacobi polynomials to which is added an arbitrary mass located at the point $x=0$, the middle of the orthogonality interval. This provides the first nontrivial example of Krall-type polynomials with a point mass inside the orthogonality interval. These polynomials can be obtained by a Geronimus transform of the little $q$-Jacobi polynomials in the limit $q=-1$.

math.CA↗

arXiv2015

On an integrable system related to the relativistic Toda lattice -Bäcklund transformation and integrable discretization

We study an integrable system related to the relativistic Toda lattice. The bilinear representation of this lattice is given and the Bäckulund transformation obtained. A fully discrete version is also introduced with its bilinear Bäcklund transformation and Lax pair. One-soliton solution of the discrete system is presented by use of Bäcklund transformation.

nlin.SI↗

arXiv2015

The equitable presentation of $\mathfrak{osp}_q(1|2)$ and a $q$-analog of the Bannai-Ito algebra

The equitable presentation of the quantum superalgebra $\mathfrak{osp}_q(1|2)$, in which all generators appear on an equal footing, is exhibited. It is observed that in their equitable presentations, the quantum algebras $\mathfrak{osp}_q(1|2)$ and $\mathfrak{sl}_q(2)$ are related to one another by the formal transformation $q\rightarrow -q$. A $q$-analog of the Bannai-Ito algebra is shown to arise as the covariance algebra of $\mathfrak{osp}_q(1|2)$.

math.QA↗

arXiv2015

The quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials

The Racah problem for the quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ is considered. The intermediate Casimir operators are shown to realize a $q$-deformation of the Bannai-Ito algebra. The Racah coefficients of $\mathfrak{osp}_q(1|2)$ are calculated explicitly in terms of basic orthogonal polynomials that $q$-generalize the Bannai-Ito polynomials. The relation between these $q$-deformed Bannai-Ito polynomials and the $q$-Racah/Askey-Wilson polynomials is discussed.

math.QA↗

arXiv2015

A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra

The Dirac-Dunkl operator on the 2-sphere associated to the $\mathbb{Z}_2^3$ reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.

math-ph↗

arXiv2015

The multivariate Hahn polynomials and the singular oscillator

Karlin and McGregor's d-variable Hahn polynomials are shown to arise in the (d+1)-dimensional singular oscillator model as the overlap coefficients between bases associated to the separation of variables in Cartesian and hyperspherical coordinates. These polynomials in d discrete variables depend on d+1 real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution. The focus is put on the d=2 case for which the connection with the three-dimensional singular oscillator is used to derive the main properties of the polynomials: forward/backward shift operators, orthogonality relation, generating function, recurrence relations, bispectrality (difference equations) and explicit expression in terms of the univariate Hahn polynomials. The extension of these results to an arbitrary number of variables is presented at the end of the paper.

math-ph↗

arXiv2014

The Generic Superintegrable System on the 3-Sphere and the $9j$ Symbols of $\mathfrak{su}(1,1)$

The $9j$ symbols of $\mathfrak{su}(1,1)$ are studied within the framework of the generic superintegrable system on the 3-sphere. The canonical bases corresponding to the binary coupling schemes of four $\mathfrak{su}(1,1)$ representations are constructed explicitly in terms of Jacobi polynomials and are seen to correspond to the separation of variables in different cylindrical coordinate systems. A triple integral expression for the $9j$ coefficients exhibiting their symmetries is derived. A double integral formula is obtained by extending the model to the complex three-sphere and taking the complex radius to zero. The explicit expression for the vacuum coefficients is given. Raising and lowering operators are constructed and are used to recover the relations between contiguous coefficients. It is seen that the $9j$ symbols can be expressed as the product of the vacuum coefficients and a rational function. The recurrence relations and the difference equations satisfied by the $9j$ coefficients are derived.

math-ph↗

arXiv2014

Two-variable $-1$ Jacobi polynomials

A two-variable generalization of the Big $-1$ Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big $-1$ Jacobi polynomials. Their orthogonality measure is obtained. Their bispectral properties (eigenvalue equations and recurrence relations) are determined through a limiting process from the two-variable Big $q$-Jacobi polynomials of Lewanowicz and Woźny. An alternative derivation of the weight function using Pearson-type equations is presented.

math.CA↗

arXiv2014

$q$-Rotations and Krawtchouk polynomials

An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown to arise as matrix elements of unitary "$q$-rotation" operators expressed as $q$-exponentials in the $\mathcal{U}_{q}(sl_{2})$ generators. The properties of the polynomials (orthogonality relation, generating function, structure relations, recurrence relation, difference equation) are derived by exploiting the algebraic setting. The results are extended to another family of polynomials, the affine $q$-Krawtchouk polynomials, through a duality relation.

math-ph↗

arXiv2014

The Bannai-Ito algebra and some applications

The Bannai-Ito algebra is presented together with some of its applications. Its relations with the Bannai-Ito polynomials, the Racah problem for the $sl_{-1}(2)$ algebra, a superintegrable model with reflections and a Dirac-Dunkl equation on the 2-sphere are surveyed.

math-ph↗

arXiv2014

Spin lattices, state transfer and bivariate Krawtchouk polynomials

The quantum state transfer properties of a class of two-dimensional spin lattices on a triangular domain are investigated. Systems for which the 1-excitation dynamics is exactly solvable are identified. The exact solutions are expressed in terms of the bivariate Krawtchouk polynomials that arise as matrix elements of the unitary representations of the rotation group on the states of the three-dimensional harmonic oscillator.

math-ph↗

arXiv2014

The multivariate Meixner polynomials as matrix elements of $SO(d,1)$ representations on oscillator states

The multivariate Meixner polynomials are shown to arise as matrix elements of unitary representations of the $SO(d,1)$ group on oscillator states. These polynomials depend on $d$ discrete variables and are orthogonal with respect to the negative multinomial distribution. The emphasis is put on the bivariate case for which the SO(2,1) connection is used to derive the main properties of the polynomials: orthogonality relation, raising/lowering relations, generating function, recurrence relations and difference equations as well as explicit expressions in terms of standard (univariate) Krawtchouk and Meixner polynomials. It is explained how these results generalize directly to $d$ variables.

math-ph↗

arXiv2014

The Dunkl-Coulomb problem in the plane

The Dunkl-Coulomb system in the plane is considered. The model is defined in terms of the Dunkl Laplacian, which involves reflection operators, with a $r^{-1}$ potential. The system is shown to be maximally superintegrable and exactly solvable. The spectrum of the Hamiltonian is derived algebraically using a realization of $\mathfrak{so}(2,1)$ in terms of Dunkl operators. The symmetry operators generalizing the Runge-Lenz vector are constructed. On eigenspaces of fixed energy, the invariance algebra they generate is seen to correspond to a deformation of $\mathfrak{su}(2)$ by reflections. The exact solutions are given as products of Laguerre polynomials and Dunkl harmonics on the circle.

math-ph↗

arXiv2014

The Dunkl oscillator in three dimensions

The isotropic Dunkl oscillator model in three-dimensional Euclidean space is considered. The system is shown to be maximally superintegrable and its symmetries are obtained by the Schwinger construction using the raising/lowering operators of the dynamical sl_{-1}(2) algebra of the one-dimensional Dunkl oscillator. The invariance algebra generated by the constants of motion, an extension of u(3) with reflections, is called the Schwinger-Dunkl algebra sd(3). The system is shown to admit separation of variables in Cartesian, polar (cylindrical) and spherical coordinates and the corresponding separated solutions are expressed in terms of generalized Hermite, Laguerre and Jacobi polynomials.

math-ph↗

arXiv2014

The multivariate Charlier polynomials as matrix elements of the Euclidean group representation on oscillator states

A family of multivariate orthogonal polynomials generalizing the standard (univariate) Charlier polynomials is shown to arise in the matrix elements of the unitary representation of the Euclidean group E(d) on oscillator states. These polynomials in d discrete variables are orthogonal on the product of d Poisson distributions. The accent is put on the d=2 case and the group theoretical setting is used to obtain the main properties of the polynomials: orthogonality and recurrence relations, difference equation, raising/lowering relations, generating function, hypergeometric and integral representations and explicit expression in terms of standard Charlier and Krawtchouk polynomials. The approach is seen to extend straightforwardly to an arbitrary number of variables. The contraction of SO(3) to E(2) is used to show that the bivariate Charlier polynomials correspond to a limit of the bivariate Krawtchouk polynomials.

math-ph↗

arXiv2014

The Bannai-Ito algebra and a superintegrable system with reflections on the 2-sphere

A quantum superintegrable model with reflections on the 2-sphere is introduced. Its two algebraically independent constants of motion generate a central extension of the Bannai--Ito algebra. The Schrodinger equation separates in spherical coordinates and its exact solutions are presented. It is further observed that the Hamiltonian of the system arises in the addition of three representations of the sl_{-1}(2) algebra (the dynamical algebra of the one-dimensional parabosonic oscillator). The contraction from the two-sphere to the Euclidean plane yields the Dunkl oscillator in two dimensions and its Schwinger-Dunkl symmetry algebra sd(2).

math-ph↗

arXiv2014

A "Continuous" Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials

A novel family of $-1$ orthogonal polynomials called the Chihara polynomials is characterized. The polynomials are obtained from a "continuous" limit of the complementary Bannai-Ito polynomials, which are the kernel partners of the Bannai-Ito polynomials. The three-term recurrence relation and the explicit expression in terms of Gauss hypergeometric functions are obtained through a limit process. A one-parameter family of second-order differential Dunkl operators having these polynomials as eigenfunctions is also exhibited. The quadratic algebra with involution encoding this bispectrality is obtained. The orthogonality measure is derived in two different ways: by using Chihara's method for kernel polynomials and, by obtaining the symmetry factor for the one-parameter family of Dunkl operators. It is shown that the polynomials are related to the big $-1$ Jacobi polynomials by a Christoffel transformation and that they can be obtained from the big $q$-Jacobi by a $q\rightarrow -1$ limit. The generalized Gegenbauer/Hermite polynomials are respectively seen to be special/limiting cases of the Chihara polynomials. A one-parameter extension of the generalized Hermite polynomials is proposed.

math.CA↗

arXiv2014

The Bannai-Ito polynomials as Racah coefficients of the sl_{-1}(2) algebra

The Bannai-Ito polynomials are shown to arise as Racah coefficients for sl_{-1}(2). This Hopf algebra has four generators including an involution and is defined with both commutation and anticommutation relations. It is also equivalent to the parabosonic oscillator algebra. The coproduct is used to show that the Bannai-Ito algebra acts as the hidden symmetry algebra of the Racah problem for sl_{-1}(2). The Racah coefficients are recovered from a related Leonard pair.

math-ph↗