  We consider a realization of fractional supersymmetric of quantum mechanics
of order $r$, where the Hamiltonian and supercharges involve reflection
operators. It is shown that the Hamiltonian has $r$-fold degenerate spectrum
and the eigenvalues of hermitian supercharges are zeros of the associated
Hermite polynomials of Askey and Wimp. Also it is shown that the associated
eigenfunctions involve lacunary Hermite polynomials.
