The $q$-calculus for generic $q$ and $q$ a root of unity
  The $q$-calculus for generic $q$ is developed and related to the deformed oscillator of parameter $q^{1/2}$. By passing with care to the limit in which $q$ is a root of unity, one uncovers the full algebraic structure of ${{\cal Z}}_n$-graded fractional supersymmetry and its natural representation. 
R. S. Dunne, A. J. Macfarlane, J. A. de Azc\'arraga and J. C. P\'erez   Bueno