**The structure relation for Askey-Wilson polynomials,
Zhedanov's algebra AW(3), and a double affine Hecke algebra**

**Abstract:**

Several authors have studied raising, lowwering and structure relations for
families of orthogonal polynomials in the Askey and *q*-Askey scheme. The
approaches are usually on a case-by-case basis, and quite computational.
However, a quick and more universal approach [1] is implicit in Zhedanov's [4]
algebra *AW*(3). This will be discussed in the first part of the lecture.

It will turn out [2] that the algebra *AW*(3) is closely related to the
double affine Hecke algebra (DAHA, [3]) for the Askey-Wilson
polynomials. This will be the main topic of the lecture.
To be a little more specific, consider *AW*(3) 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. 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 DAHA related to symmetric and
non-symmetric Askey-Wilson polynomials will be presented without
requiring knowledge of general DAHA theory. Finally a central
extension of this quotient of *AW*(3) will be introduced which can be
embedded in the DAHA by means of the faithful basic representations of
both algebras.

[1] T.H. Koornwinder, The structure relation for Askey-Wilson polynomials,
*J. Comput. Appl. Math.*, in press; math.CA/0601303.

[2] T.H. Koornwinder, The relationship between Zhedanov's algebra
*AW*(3) and the double affine Hecke algebra in the rank one case,
*SIGMA* **3** (2007), 063; math.QA/0612730.

[3] I.G. Macdonald, Affine Hecke algebra and orthogonal polynomials,
Cambridge University Press, 2003.

[4] A.S. Zhedanov, "Hidden symmetry" of Askey-Wilson polynomials,
*Theoret. and Math. Phys.* **89** (1991), 1146-1157.