On the structure and probabilistic interpretation of Askey–Wilson densities and polynomials with complex parameters
AbstractWe 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’.
|Journal series||Journal of Functional Analysis, ISSN 0022-1236, (A 40 pkt)|
|Keywords in English||Al-Salam–Chihara polynomials, Askey–Wilson density, Askey–Wilson polynomials, Poisson–Mehler expansion, q-Hermite polynomials|
|Publication indicators||: 2011 = 1.082 (2) - 2011=1.208 (5)|
|Citation count*||6 (2015-04-21)|
* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perish system.