Multiple Orthogonal Polynomials on the Ball and Radial Extensions

A primary method for constructing orthogonal polynomials on the unit ball consists of combining a Jacobi-type radial component with a spherical harmonic angular part. Building upon this framework and using Jacobi-Pi\~neiro multiple orthogonal polynomials, this paper introduces Type I and Type II multiple orthogonal polynomials on the multidimensional ball. To demonstrate the practical utility of these definitions, we establish multivariate extensions of several fundamental results from univariate multiple orthogonality. Finally, we extend the construction to more general domains by introducing multiple orthogonality with respect to radial weights.