24 citations · 158 across the 23 of their papers we have counts for
24 papers
Change of basis for the tridiagonal pairs of type II
Nicolas Crampe, Julien Gaboriaud, Satoshi Tsujimoto
We study tridiagonal pairs of type II. These involve two linear transformations and . We define two bases. In the first one, acts as a diagonal matrix while $A^\st…
-deformed Griffiths polynomials of Racah type
Nicolas Crampe, Luc Frappat, Julien Gaboriaud +1
New bivariate Griffiths polynomials of -Racah type are introduced and characterized. They generalize the polynomials orthogonal on the multinomial distribution introduced by R.…
Griffiths polynomials of Racah type
Nicolas Crampe, Luc Frappat, Julien Gaboriaud +3
Bivariate Griffiths polynomials of Racah type are constructed from univariate Racah polynomials. The bispectral properties of the former are deduced from simple properties of the l…
-Griffiths polynomials: Bispectrality and biorthogonality
N. Crampe, L. Frappat, J. Gaboriaud +3
We introduce a generalization of bivariate Griffiths polynomials depending on an additional parameter . These -Griffiths polynomials are bivariate, bispectral and biorthogona…
Matrix elements of in representations as bispectral multivariate functions
Nicolas Crampe, Julien Gaboriaud, Loïc Poulain d'Andecy +1
We compute the matrix elements of in any finite-dimensional irreducible representation of . They are expressed in terms of a double sum of products of Krawtchouk and…
Bispectrality and biorthogonality of the rational functions of -Hahn type
Ismaël Bussière, Julien Gaboriaud, Luc Vinet +1
We introduce families of rational functions that are biorthogonal with respect to the -hypergeometric distribution. A triplet of -difference operators , , is shown…
Racah algebras, the centralizer and its Hilbert-Poincaré series
Nicolas Crampe, Julien Gaboriaud, Loïc Poulain d'Andecy +1
The higher rank Racah algebra introduced recently is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra , is then intro…
Orthogonal polynomials and the deformed Jordan plane
André Beaudoin, Geoffroy Bergeron, Antoine Brillant +3
We consider the unital associative algebra with two generators , obeying the defining relation .…
The rational Sklyanin algebra and the Wilson and para-Racah polynomials
Geoffroy Bergeron, Julien Gaboriaud, Luc Vinet +1
The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grid…
The Askey-Wilson algebra and its avatars
Nicolas Crampé, Luc Frappat, Julien Gaboriaud +3
The original Askey-Wilson algebra introduced by Zhedanov encodes the bispectrality properties of the eponym polynomials. The name 'Askey-Wilson algebra' is currently used to refer…
Sklyanin-like algebras for (-)linear grids and (-)para-Krawtchouk polynomials
Geoffroy Bergeron, Julien Gaboriaud, Luc Vinet +1
S-Heun operators on linear and -linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The Continuous Hahn a…
Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators
Julien Gaboriaud, Satoshi Tsujimoto, Luc Vinet +1
The -difference equation, the shift and the contiguity relations of the Askey-Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin…
A Howe correspondence for the algebra of the Clebsch-Gordan coefficients
Julien Gaboriaud, Luc Vinet
Two descriptions of the dual Hahn algebra are presented and shown to be related under Howe duality. The dual pair involved is formed by the Lie algebra and t…
Superintegrability and the dual Hahn algebra in superconformal quantum mechanics
Pierre-Antoine Bernard, Julien Gaboriaud, Luc Vinet
A two-dimensional superintegrable system of singular oscillators with internal degrees of freedom is identified and exactly solved. Its symmetry algebra is seen to be the dual …
Howe duality and algebras of the Askey-Wilson type: an overview
Julien Gaboriaud, Luc Vinet, Stéphane Vinet
The Askey-Wilson algebra and its relatives such as the Racah and Bannai-Ito algebras were initially introduced in connection with the eponym orthogonal polynomials. They have since…
Revisiting the Askey--Wilson algebra with the universal R-matrix of
Nicolas Crampe, Julien Gaboriaud, Luc Vinet +1
A description of the embedding of the universal Askey--Wilson algebra, AW(3), in is given in terms of the universal R-matrix of . The generators…
The dual pair , -oscillators and Askey-Wilson algebras
Luc Frappat, Julien Gaboriaud, Eric Ragoucy +1
The universal Askey-Wilson algebra can be obtained as the commutant of in . We analyze the commutant of $\math…
The -Higgs and Askey-Wilson algebras
Luc Frappat, Julien Gaboriaud, Eric Ragoucy +1
A -analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the -sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2…
The Higgs and Hahn algebras from a Howe duality perspective
Luc Frappat, Julien Gaboriaud, Luc Vinet +2
The Hahn algebra encodes the bispectral properties of the eponymous orthogonal polynomials. In the discrete case, it is isomorphic to the polynomial algebra identified by Higgs as…
The dual pair , the Dirac equation and the Bannai-Ito algebra
Julien Gaboriaud, Luc Vinet, Stéphane Vinet +1
The Bannai-Ito algebra can be defined as the centralizer of the coproduct embedding of in . It will be shown that it is also…
The generalized Racah algebra as a commutant
Julien Gaboriaud, Luc Vinet, Stéphane Vinet +1
The Racah algebra of rank is obtained as the commutant of the \mbox{} subalgebra of in oscillator representations of t…
The Racah algebra as a commutant and Howe duality
Julien Gaboriaud, Luc Vinet, Stéphane Vinet +1
The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the -sphere. It is further…
An algebraic interpretation of the -Meixner polynomials
Julien Gaboriaud, Luc Vinet
An algebraic interpretation of the -Meixner polynomials is obtained. It is based on representations of on -oscillator states with the poly…
A superintegrable discrete oscillator and two-variable Meixner polynomials
Julien Gaboriaud, Vincent X. Genest, Jessica Lemieux +1
A superintegrable, discrete model of the quantum isotropic oscillator in two-dimensions is introduced. The system is defined on the regular, infinite-dimensional $\mathbb{N}\times…