A method for generating multivariate multiple orthogonal polynomials
In the framework of multiple orthogonal polynomials (MOPs) and their extension to the multivariate setting, a unified mechanism for generating structured families of multiple orthogonal polynomials across diverse multidimensional domains is still lacking. In this work, we bridge this gap by developing an extended Koornwinder-type methodology for constructing multivariate multiple orthogonal polynomials. We introduce two distinct coupling schemes: combining a univariate MOPs family with a standard orthogonal family, which enables the definition of both Type I and Type II bivariate systems along with their dual biorthogonality relations, and coupling two univariate MOPs families to form Type II multivariate systems. Using this general framework, we provide the first explicit formulations of multiple orthogonal polynomials on a variety of bivariate regions, including bounded domains such as a parabolic domain and the square $[0,1]^2$, non-standard unbounded geometries like the positive quadrant $(\mathbb{R}_0^+)^2$ and the wedge $\mathbb{V}^2$, as well as new constructions on the triangle $T$. Furthermore, explicit multiple orthogonal systems on the $d$-dimensional simplex $T^d$ and on the $d$-dimensional cone $\mathbb{V}^d$ are given.