Algebraic shape invariant potentials as the generalized deformed oscillator

Within the framework of supersymmetric quantum mechanics, we study the simplified version of the potential algebra of the shape invariance condition in k steps, where k is an arbitrary positive integer. The associated potential algebra is found to be equivalent to the generalized deformed oscillator algebra that has a built-in Z(k)-grading structure. The algebraic realization of the shape invariance condition in k steps is therefore formulated by the method of the Z(k)-graded deformed oscillator. Based on this formulation, we explicitly construct the general algebraic properties for shape invariant potentials in k steps, in which the parameters of partner potentials are related to each other by the translation a(1) = a(0) + delta. The obtained results include the cyclic shape invariant potentials of period k as a special case.