Wick calculus for noncommutative white noise corresponding to $q$-deformed commutation relations

We derive the Wick calculus for test and generalized functionals of noncommutative white noise corresponding to $q$-deformed commutation relations with $q\in(-1,1)$. We construct a Gel'fand triple centered at the $q$-deformed Fock space in which both the test, nuclear space and its dual space are algebras with respect to the addition and the Wick multiplication. Furthermore, we prove a V\r{a}ge-type inequality for the Wick product on the dual space.