Hermite and Laguerre \(2D\) polynomials (Q5946636)

The Hermite \(2D\) polynomials \(H_{m,n} (U;x,y)\) and Laguerre \(2D\) polynomials \(L_{m,n} (U;z,\overline z)\) are defined as functions of two variables with an arbitrary \(2D\) matrix \(U\) as parameter. Their properties are discussed, explicit representations are given and recursion relations and generating functions for these polynomials are derived. Of practical importance is the fact that the Hermite and Laguerre \(2D\) polynomials satisfy in comparison to the related usual two-variable Hermite polynomials orthogonality relations in a direct way.