Recursive formulas for the overlaps between Bethe states and product states in XXZ Heisenberg chains

We consider the problem of computing the overlaps between the Bethe states of the XXZ spin-1/2 chain and generic states. We derive recursive formulas for the overlaps between some simple product states and off-shell Bethe states within the framework of the Algebraic Bethe Ansatz. These recursive formulas can be used to prove in a simple and straightforward way the recently-obtained results for the overlaps of the Bethe states with the N\'eel state, the dimer state, and the \textit{q}-deformed dimer state. However, these recursive formulas are derived for a broader class of states and represent a concrete starting point for the computation of rather general overlaps. Our approach can be easily extended to other one-dimensional Bethe Ansatz integrable models.