COVARIANT Q-DIFFERENTIAL CALCULUS AND ITS DEFORMATIONS AT Q(N)=1

We construct the generalized version of covariant Z(3)-graded differen tial calculus introduced by one of us (R.K.) and then extend it to the case of arbitrary Z(N) grading. Here our main purpose is to establish the recurrence formulae for the Nth power of covariant q-differential D-q = d(q) + A and to analyze more closely the particular case of q b eing an Nth primitive root of unity. The generalized notions of connec tion and curvature are introduced and several examples of realization are displayed for N = 3 and 4. Finally we briefly discuss the idea of infinitesimal deformations of the parameter q in the complex plane.