2010/11/30 by Paweł J. Szabłowski · 1 citation
Computer Science · Mathematics · #Askey–Wilson polynomials #Bayesian Methods and Mixture Models #Discrete orthogonal polynomials #Hermite polynomials #Interpretation (philosophy) #Mathematical functions and polynomials #Mathematics #Orthogonal polynomials #Poisson distribution #Probabilistic logic #Probability density function #Pure mathematics #Statistical Distribution Estimation and Applications #Statistics #Wilson polynomials #math.CA #math.PR #msc:05A30 #msc:26C05 #msc:33D45 #msc:33D50 #msc:42A16 #msc:60E05 #msc:62H05
paper · pdf · doi:10.1016/j.jfa.2011.04.002
published as J. Funct. Anal. 261 (2011), no. 3, 635--659
openalex publication_date 2011/04/24 · arxiv created 2011/05/03 · arxiv updated 2012/08/13 · openalex created_date 2017/10/20 · openalex updated_date 2026/08/05
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'.