Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle

Abstract

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.

References in corpus (8)

• The Universal Askey-Wilson Algebra
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Double Affine Hecke Algebras of Rank 1 and the $Z_3$-Symmetric Askey-Wilson Relations
• Raising and Lowering Operators for Askey-Wilson Polynomials
• Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs
• The universal DAHA of type $(C_1^{\vee },C_1)$ and Leonard pairs of $q$-Racah type
• Finite dimensional representations of double affine Hecke algebra of rank 1
• CMV matrices and little and big -1 Jacobi polynomials