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 one-dimensional conditional distribution and then checking Chapman–Kolmogorov equation. Support of this distribution consists 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 [Formula: see text], where Hk(x) denotes kth Hermite polynomial.