q-Orthogonal dualities for asymmetric particle systems

We study a class of interacting particle systems with asymmetric interaction showing a self-duality property. The class includes the ASEP($q,\theta$), asymmetric exclusion process, with a repulsive interaction, allowing up to $\theta\in \N$ particles in each site, and the ASIP$(q,\theta)$, $\theta\in \R^+$, asymmetric inclusion process, that is its attractive counterpart. We extend to the asymmetric setting the investigation of orthogonal duality properties done in \cite{CFGGR} for symmetric processes. The analysis leads to {multivariate} $q-$analogues of {Krawtchouk} polynomials and Meixner polynomials as orthogonal duality functions for the generalized asymmetric exclusion process and its asymmetric inclusion version, respectively.