Refining the Shifted Topological Vertex

We study aspects of the refining and shifting properties of the 3d MacMahon function $\mathcal{C}_{3}(q) $ used in topological string theory and BKP hierarchy. We derive the explicit expressions of the shifted topological vertex $\mathcal{S}_{\lambda \mu \nu}(q) $ and its refined version $\mathcal{T}_{\lambda \mu \nu}(q,t) $. These vertices complete results in literature.