F-polynomials in Quantum Cluster Algebras

F -polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra. In this paper, we define and prove the existence of analogous quantum F -polynomials for quantum cluster algebras. We prove some properties of quantum F -polynomials. In particular, we give a recurrence relation which can be used to compute them. Finally, we compute quantum F -polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type An quantum cluster algebras.