Hermite and Laguerre 2D polynomials

We define Hermite 2D polynomials H-m,H-n(U;x,y) and Laguerre 2D polynomials L-m,L-n(U;z,(z) over bar) as functions of two variables with an arbitrary 2D matrix U as parameter and discuss their properties and their explicit representation. Recursion relations and generating functions for these polynomials are derived. The advantage of the introduced Hermite and Laguerre 2D polynomials in comparison to the related usual two-variable Hermite polynomials is that they satisfy orthogonality relations in a direct way, whereas for the purpose of orthonormalization of the last, one has to introduce two different kinds of such polynomials which are biorthogonal to each other. (C) 2001 Elsevier Science B.V. All rights reserved.