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.