On the structure and probabilistic interpretation of Askey–Wilson densities and polynomials with complex parameters

We give equivalent forms of the Askey–Wilson polynomials expressing them with the help of the Al-Salam–Chihara polynomials. After restricting parameters of the Askey–Wilson polynomials to complex conjugate pairs we expand the Askey–Wilson weight function in the series similar to the Poisson–Mehler expansion formula and give its probabilistic interpretation. In particular this result can be used to calculate explicit forms of ‘q-Hermite’ moments of the Askey–Wilson density, hence enabling calculation of all moments of the Askey–Wilson density. On the way (by setting certain parameter q to 0) we get some formulae useful in the rapidly developing so-called ‘free probability’.