Befriending Askey-Wilson polynomials

Abstract

We recall five families of polynomials constituting a part of the so-called Askey-Wilson scheme. We do this to expose properties of the Askey-Wilson (AW) polynomials that constitute the last, most complicated element of this scheme. In doing so we express AW density as a product of the density that makes $q-$Hermite polynomials orthogonal times a product of four characteristic function of $q-$Hermite polynomials (\ref{fAW}) just pawing the way to a generalization of AW integral. Our main results concentrate mostly on the complex parameters case forming conjugate pairs. We present new fascinating symmetries between the variables and some newly defined (by the appropriate conjugate pair) parameters. In particular in (\ref% {rozwiniecie1}) we generalize substantially famous Poisson-Mehler expansion formula (\ref{PM}) in which $q-$Hermite polynomials are replaced by Al-Salam-Chihara polynomials. Further we express Askey-Wilson polynomials as linear combinations of Al-Salam-Chihara (ASC) polynomials. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we reach AW polynomials and AW densities that have clear probabilistic interpretation.

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References in corpus (5)

• Conditional moments of q-Meixner processes
• Expansions of one density via polynomials orthogonal with respect to the other
• On the structure and probabilistic interpretation of Askey-Wilson densities and polynomials with complex parameters
• Towards a q-analogue of the Kibble--Slepian formula in 3 dimensions
• On summable, positive Poisson-Mehler kernels built of Al-Salam--Chihara and related polynomials

Cited by in corpus (5)

• Moments of q-Normal and conditional q-Normal distributions
• On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey--Wilson, Al-Salam--Chihara and q-ultraspherical polynomials
• On positivity of orthogonal series and its applications in probability
• On the families of polynomials forming a part of the Askey--Wilson scheme and their probabilistic applications
• On the generalized Kesten--McKay distributions