  Generalised matrix elements of the irreducible representations of the quantum
$SU(2)$ group are defined using certain orthonormal bases of the representation
space. The generalised matrix elements are relatively infinitesimal invariant
with respect to Lie algebra like elements of the quantised universal enveloping
algebra of $sl(2)$. A full proof of the theorem announced by Noumi and Mimachi
[Proc. Japan Acad. Sci. {\bf 66}, Ser. A (1990), pp. 146--149] describing the
generalised matrix elements in terms of the full four-parameter family of
Askey-Wilson polynomials is given. Various known and new applications of this
interpretation are presented.
