The role of Jacobi polynomials in the theory of Hermite and Laguerre 2D polynomials

Using an alternative definition of usual Hermite polynomials, two problems in the theory of general Hermite and Laguerre 2D polynomials can be separated with advantage for the further treatment: the introduction of a general 2D matrix in the linear transformation of powers of the components of a 2D vector and the generation of Hermite (or Laguerre) polynomials by applying an integral operator to these powers. The Jacobi polynomials appear in the finite-dimensional irreducible representations of the two-dimensional general linear group GL(2,C).