  In this paper we introduce a basic representation for the confluent Cherednik
algebras $\mathcal H_{\rm V}$, $\mathcal H_{\rm III}$, $\mathcal H_{\rm
III}^{D_7}$ and $\mathcal H_{\rm III}^{D_8}$ defined in arXiv:1307.6140. To
prove faithfulness of this basic representation, we introduce the non-symmetric
versions of the continuous dual $q$-Hahn, Al-Salam-Chihara, continuous big
$q$-Hermite and continuous $q$-Hermite polynomials.
