Rectangular Recurrence Relations in ${\mathfrak{g}\mathfrak{l}}_n$ and ${\mathfrak{o}}_{2n+1}$ Invariant Integrable Models

A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type ${\mathfrak{g}\mathfrak{l}}_n$ or type ${\mathfrak{o}}_{2n+1}$ symmetries. These recurrence relations describe how to add a single parameter $z$ to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on $z$. We refer to these recurrence relations as rectangular because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method.