  The discrete Fourier analysis on the
$30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the
corresponding results on the regular hexagon by considering functions invariant
under the group $G_2$, which leads to the definition of four families
generalized Chebyshev polynomials. The study of these polynomials leads to a
Sturm-Liouville eigenvalue problem that contains two parameters, whose
solutions are analogues of the Jacobi polynomials. Under a concept of
$m$-degree and by introducing a new ordering among monomials, these polynomials
are shown to share properties of the ordinary orthogonal polynomials. In
particular, their common zeros generate cubature rules of Gauss type.
