Related papers: Dualities in the $q$-Askey scheme and degenerate D…
Zhedanov's algebra AW(3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this…
We study Askey-Wilson type polynomials using representation theory of the double affine Hecke algebra. In particular, we prove bi-orthogonality relations for non-symmetric and anti-symmetric Askey-Wilson polynomials with respect to a…
This paper builds on the previous paper arXiv:math/0612730 by the author, where a relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was established. It is…
In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\check{C_1}C_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the…
The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey-Wilson and q-Racah polynomials. As it is well…
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 to a variety of related structures that appear in a large…
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 construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper…
An algebra is introduced which can be considered as a rank 2 extension of the Askey-Wilson algebra. Relations in this algebra are motivated by relations between coproducts of twisted primitive elements in the two-fold tensor product of the…
In this paper, by introducing new matrix operations and using a specific inverse relation, we establish the dual forms of the orthogonality relations for some well-known discrete and continuous $q$-orthogonal polynomials from the…
The $q$-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 algebras $\mathfrak{ska}_3$ and $\mathfrak{ska}_4$. It is…
This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the $q$-Askey scheme and uniform ways of parametrizing the families. For a system of polynomials $p_n(x)$ in the…
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 proved ubiquitous. In particular they admit presentations…
An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to…
In his famous theorem (1982), Douglas Leonard characterized the $q$-Racah polynomials and their relatives in the Askey scheme from the duality property of $Q$-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or…
Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum mechanical formulation, idQM (discrete quantum mechanics…
The Askey-Wilson polynomials are orthogonal polynomials in $x = \cos \theta$, which are given as a terminating $_4\phi_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{i\theta}$,…
A new trigonometric degeneration of the Sklyanin algebra is found and the functional realization of its representations in space of polynomials in one variable is studied. A further contraction gives the standard quantum algebra…
The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based…
Assume that $\mathbb F$ is an algebraically closed field and let $q$ denote a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra defined…