2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59077We construct the generalized version of covariant Z_3-graded differential calculus introduced by one of us (R.K.), and then extended to the case of arbitrary Z_N grading. Here our main purpose is to establish the recurrence formulae for the N-th power of covariant q-differential D_q = d_q + A and to analyze more closely the particular case of q being an Nth primitive root of unity. The generalized notions of connection and curvature are introduced and several examples of realization are displayed for N=3 and N=4. Finally we briefly discuss the idea of infinitesimal deformations of the parameter q in the complex plane.16 pages, no figuresQuantum AlgebraDifferential Geometry53-XX, 15-XX, 81-XXtext