2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219043We 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}_{λμν}(q) $ and its refined version $\mathcal{T}_{λμν}(q,t) $. These vertices complete results in literature.Latex, 14 pages, 2 figures. To appear in Jour Math PhysHigh Energy Physics - Theorytext