Jacobi-Angelesco multiple orthogonal polynomials on an $r$-star

We investigate type I multiple orthogonal polynomials on $r$ intervals which have a common point at the origin and endpoints at the $r$ roots of unity ${\omega }^j$, $j=0,1,\dots ,r-1$, with $\omega =\exp (2\pi i\mathrm{/}r)$. We use the weight function $\mathrm{\mid }x{\mathrm{\mid }}^{\beta }(1-x^r{)}^{\alpha }$, with $\alpha ,\beta >-1$ for the multiple orthogonality relations. We give explicit formulas for the type I multiple orthogonal polynomials, the coefficients in the recurrence relation, the differential equation, and we obtain the asymptotic distribution of the zeros.