Askey--Wilson Integral and its Generalizations

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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.