# Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle

@article{Tsujimoto2019DoubleAH,
title={Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle},
author={Satoshi Tsujimoto and Luc Vinet and Alexei S. Zhedanov},
journal={Constructive Approximation},
year={2019}
}
• Published 21 September 2017
• Mathematics
• Constructive Approximation
An inifinite-dimensional representation of the double affine Hecke algebra of rank 1 and type $(C_1^{\vee},C_1)$ in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that are orthogonal on the unit circle. These polynomials can be considered as circle analogs of the Askey-Wilson polynomials. The corresponding polynomials orthogonal on an interval are constructed and discussed.
3 Citations
Dualities in the q -Askey Scheme and Degenerate DAHA
• Mathematics, Physics
Studies in Applied Mathematics
• 2018
The Askey–Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials Rn[z] that are eigenfunctions of a second-order q-difference operator L, and of a second-order
Finite-dimensional modules of the universal Askey–Wilson algebra and DAHA of type $$(C_1^\vee ,C_1)$$
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
The Askey–Wilson algebra and its avatars
• Physics, Mathematics
Journal of Physics A: Mathematical and Theoretical
• 2020
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

#### References

SHOWING 1-10 OF 49 REFERENCES
Double Affine Hecke Algebras of Rank 1 and the Z 3 -Symmetric Askey-Wilson Relations
• Mathematics
• 2010
We consider the double affine Hecke algebra H = H(k0,k1,k _ ,k _ ;q) associated with the root system (C _ 1 ,C1). We display three elements x, y, z in H that satisfy essentially the Z3-symmetric
Fourier transforms related to a root system of rank 1
We introduce an algebra $\mathcal H$ consisting of difference-reflection operators and multiplication operators that can be considered as a q = 1 analogue of Sahi's double affine Hecke algebra
Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle
• Mathematics
• 2002
It is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five-diagonal matrices depending on the Schur parameters. This result is achieved
Finite dimensional representations of the double affine Hecke algebra of rank 1
• Mathematics
• 2004
Abstract We classify the finite dimensional irreducible representations of the double affine Hecke algebra (DAHA) of type C ∨ C 1 in the case when q is not a root of unity.
Affine Hecke Algebras and Orthogonal Polynomials
Introduction 1. Affine root systems 2. The extended affine Weyl group 3. The braid group 4. The affine Hecke algebra 5. Orthogonal polynomials 6. The rank 1 case Bibliography Index.
The non-symmetric Wilson polynomials are the Bannai-Ito polynomials
• Mathematics, Physics
• 2015
The one-variable non-symmetric Wilson polynomials are shown to coincide with the Bannai-Ito polynomials. The isomorphism between the corresponding degenerate double affine Hecke algebra of type
Orthogonal polynomials and some q-beta integrals of Ramanujan☆
Abstract Two integrals of Ramanujan are used to define a q-analogue of the Euler beta integral on the real line and of the Cauchy beta-integral on the complex unit circle. Such integrals are
The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
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
CMV Matrices and Little and Big −1 Jacobi Polynomials
• Mathematics, Physics
• 2011
We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related to the theory of CMV matrices. It contains an arbitrary
On Some Classes of Polynomials Orthogonal on Arcs of the Unit Circle Connected with Symmetric Orthogonal Polynomials on an Interval
Starting from the Delsarte?Genin (DG) mapping of the symmetric orthogonal polynomials on an interval (OPI) we construct a one-parameter family of polynomials orthogonal on the unit circle (OPC). The