Probabilistic implications of symmetries of q-Hermite and Al-Salam-Chihara polynomials

Abstract

We prove the existence of stationary random fields with linear regressions for $q>1$ and thus close an open question posed by W. Bryc et al.. We prove this result by describing a discrete 1 dimensional conditional distribution and then checking Chapman-Kolmogorov equation. Support of this distribution consist of zeros of certain Al-Salam-Chihara polynomials. To find them we refer to and expose known result concerning addition of $q-$ exponential function. This leads to generalization of a well known formula $(x+y)^{n}% =\sum_{i=0}^{n}\binom{n}{k}i^{k}H_{n-k}(x) H_{k}(-iy) ,$ where $H_{k}(x) $ denotes $k-$th Hermite polynomial.