  We expand the Askey--Wilson (AW) density in a series of products of
continuous $q-$Hermite polynomials times the density that makes these
polynomials orthogonal. As a by-product we obtain the value of the AW integral
as well as the values of integrals of $q-$Hermite polynomial times the AW
density ($q-$Hermite moments of AW density). Our approach uses nice, old
formulae of Carlitz and is general enough to venture a generalization. We prove
that it is possible and pave the way how to do it. As a result we obtain system
of recurrences that if solved successfully gives a sequence of generalized AW
densities with more and more parameters.
