Related papers: Askey-Wilson polynomials: an affine Hecke algebrai…
A special Infeld-Hall factorization is given for the Askey-Wilson second order q-difference operator. It is then shown how to deducd a generalization of the corresponding Askey-Wilson polynomials.
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
We study matrix elements of a change of base between two different bases of representations of the quantum algebra $U_q(su(1,1))$. The two bases, which are multivariate versions of Al-Salam--Chihara polynomials, are eigenfunctions of…
The affine Hecke algebra of type $A$ has two parameters $\left( q,t\right) $ and acts on polynomials in $N$ variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys-Murphy…
The decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the Howe duality theorem and leads to the study of spherical harmonics.…
We develop a unified construction of matrix-valued orthogonal polynomials associated with discrete weights, yielding bispectral sequences as eigenfunctions of second-order difference operators. This general framework extends the discrete…
We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play…
We study the representation theory of the infinite type A Hecke algebra over a non-archimedean field in the case where the parameter is a pseudo-uniformizer. Specifically, we consider a family of representations, called almost-symmetric,…
In this paper all of the classical constructions of A. Young are generalized to affine Hecke algebras of type A. It is proved that the calibrated irreducible representations of the affine Hecke algebra are indexed by placed skew shapes and…
We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2\psi_2$ series, they are…
New bispectral orthogonal polynomials are obtained from an unconventional truncation of the Askey-Wilson polynomials. In the limit $q \to 1$, they reduce to the para-Racah polynomials which are orthogonal with respect to a quadratic…
These are the lecture notes of a series of lectures on Dunkl operators. We discuss the underlying algebraic structure of the degenerate double affine Hecke algebra, intertwiners and shift operators. We apply this to Macdonald theory. We…
After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts…
This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skew orthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials underlying the Askey scheme, we explicitly construct the classical skew…
We derive and study expansions of and over the Askey--Wilson polynomials. We study these expansions and examine some limits to the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite and continuous $q$-Hermite…
This is a tutorial on duality properties of special functions, mainly of orthogonal polynomials in the ($q$-)Askey scheme. It is based on the first part of the 2017 R.P. Agarwal Memorial Lecture delivered by the author.
We construct and study an explicit simultaneous $\mathscr{Y}$ eigenbasis of Ion and Wu's standard representation of the $^+$stable-limit double affine Hecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basis arises as a…
In this paper we present a unified approach to the spectral analysis of an hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor…
We prove a projection formula for the four-parameter family of orthogonal polynomials that are a reparameterization of the polynomials in the Askey-Wilson class. By carefully analyzing the recurrence relations we manage to avoid using the…
The occurrence of the Askey-Wilson (AW) algebra in the $SU(2)$ Chern-Simons (CS) theory and in the Reshetikhin-Turaev (RT) link invariant construction with quantum algebra $U_q(\mathfrak{su}_2)$ is explored. Tangle diagrams with three…