The generic-basis recurrence for regular simples in nonhomogeneous tubes

Let $Q$ be a tame quiver. Let $E$ be a regular simple module in a nonhomogeneous tube of rank $n$, and let $E(\lambda)$ be the regular simple module in the homogeneous tube for $\lambda\in\mathbb{P}^1$. Write $X_{M}$ for the quantum cluster character of a module $M$, and let $\tau_Q$ denote the Auslander–Reiten translation of $Q$.

Generic-basis recurrence. With these notations, one has

$$X_{E[n]}=X_{E(\lambda)}+q^{\frac{1}{2}}X_{(\tau_Q^{-1}E)[n-2]}.$$

This identity refines the recurrence established in the affine types considered earlier and is intended to support the construction of a generic basis for the associated quantum affine cluster algebra. Its resolution is not supplied in the source.