This is mainly a survey paper which treats the relations between three different topics: basic or \(q\)-series, the Hankel transform and certain algebras of affine Hecke type. The first two topics are classical. The basic analogues of classical functions of complex analysis are obtained by replacing a parameter \(n\) by \((1- q^n)/(1- q)\). This leads to a rich class of functions, of particular interest in number theory and combinatorics. In particular Gauss’ evaluation of the “sign” of the Gauss sum depended on a basic analogue of the binomial theorem. The authors show first of all that the theory of two types of Hankel transform, whose kernels are Bessel functions, sometimes in conjunction with a “shift operator”, give a powerful method of proving identities in the theory of basic functions. Having done this, they inquire into the formal and algebraic nature of the Hankel operator. This leads to the introduction of the Dunkl operator and the theory of two algebras of Hecke type (denoted by symbols beyond the reviewer’s skill to reproduce) and their finite-dimensional representations. These algebras are defined by generators and relations but can be given a topological interpretation and are also related to certain affine Lie algebras. The authors then treat the cases where the \(q\) is \(p\)-adic, the singular case where \(q\) is a root of unity and briefly, at the end of the paper, the limiting case where \(q\to\infty\), which leads us back to the classical Fourier transform. The determination of the finite-dimensional representations of these algebras in the cases of generic \(q\) and where \(q\) is a root of unity are the main new results of this paper.