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

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

2010-11-06

arxiv.org/abs/1011.1541v6

Mathematics

Classical Analysis and ODEs

Scientific paper

10.1016/j.jfa.2011.04.002

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'.

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