Mathematics – Functional Analysis
Scientific paper
2011-11-02
Mathematics
Functional Analysis
25
Scientific paper
Although our main interest is with the Askey-Wilson (AW) polynomials we recall and review four other families of the so-called Askey-Wilson scheme of polynomials. We do this for completeness as well as for better exposition of AW properties. Our main results concentrate on the complex parameters case, revealing new fascinating symmetries between the variables and some of the parameters. In particular we express Askey-Wilson polynomials as linear combinations of Al-Salam--Chihara (ASC) polynomials which together with the obtained earlier expansion of the Askey-Wilson density forms complete generalization of the situation met in the case of Al-Salam--Chihara and q-Hermite polynomials and the Poisson-Mehler expansion formula. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we arrive to AW polynomials and AW densities that have clear probabilistic interpretation. We recall some known and present some believed to be unknown identities and integrals.
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