On \(F\)-polynomials for generalized quantum cluster algebras and Gupta's formula (Q6614605)

The authors extend certain structure results of (quantum) cluster algebras to the setting of generalized (quantum) cluster algebras. Under a certain condition, they prove the polynomial property of \(F\)-polynomials and establish the separation formulas for generalized quantum cluster algebras. They also obtain Gupta's formulas of \(F\)-polynomials for generalized quantum cluster algebras. These formulas degenerate to Gupta's formulas of cluster algebras, but Gupta's formulas of generalized quantum cluster algebras do not degenerate to Gupta's formulas of generalized cluster algebras.