Fractional supersymmetric quantum mechanics

In this paper, two approaches of N = 2 fractional supersymmetric quantum mechanics of order k are studied in a complementary way. The first one, based on a generalized Weyl-Heisenberg algebra W-k, (that comprizes the affine quantum algebra U-q(sl(2)) with q(k) = 1 as a special case), apparently contains solely one bosonic degree of freedom. The second one uses generalized bosonic and k-fermionic degrees of freedom. As an illustration, particular emphasis is put on the fractional supersymmetric oscillator of order k.