Supersymmetric Quantum mechanics on the radial lines

We investigate a type of Hermite orthogonal polynomials on $r$ lines in the plane which have a common point at the origin and endpoints at the $r$ roots of unity and we show that their related Hermite functions are eigenfunctions of a differential-difference operator. A supersymmetric harmonic oscillator on $r$ radial lines is presented and analyzed. Its eigenfunctions are given in terms of these polynomials.