A dual and conjugate system for $q$-Gaussians for all $q$

Abstract: We provide a concrete formula for a dual system as well as for a conjugate system of $q$-Gaussians represented on the $q$-deformed Fock space. Moreover, using this formula, we prove the existence of a free Gibbs potential and that the non-microstates free Fisher information is finite for any $q$ with $-1<q<1$, which is an improvement on a previous result of Y. Dabrowski. We also indicate how our results can be extended to the more general setting of mixed $q_{ij}$-relations.