Fractional supersymmetric quantum mechanics and lacunary Hermite polynomials

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.