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 $ω^j$, $j=0,1,\ldots,r-1$, with $ω= \exp(2πi/r)$. We use the weight function $|x|^β(1-x^r)^α$, with $α,β>-1$ 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.

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