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 Askey–Wilson polynomials. It is proved that this representation is faithful for a certain quotient of $AW(3)$ such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to symmetric and non-symmetric Askey–Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of $AW(3)$ is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.