On 𝐹-Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula

Abstract

We show the polynomial property of 𝐹-polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of 𝐹-polynomials for generalized quantum cluster algebras. These formulas specialize to Gupta's formulas for quantum cluster algebras and cluster algebras, respectively. Finally, a generalization of Gupta's formula has also been discussed in the setting of generalized cluster algebras.