  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.
