Raising and Lowering Operators for Askey–Wilson Polynomials

In this paper we describe two pairs of raising/lowering operators for Askey–Wilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the “classical” properties of these polynomials, viz. the $q$-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra.