We revisit the shift technique applied to Quasi-Birth and Death (QBD) processes He et al. (2001) [13] in functional form by bringing the attention to the existence and properties of canonical factorizations. To this regard, we prove new results concerning the solutions of the quadratic matrix equations associated with the QBD. These results find applications to the solution of the Poisson equation for QBDs. (c) 2016 IMACS. Published by Elsevier B.V. All rights reserved.