General Hermite and Laguerre two-dimensional (2D) polynomials which form a
(complex) three-parameter unification of the special Hermite and Laguerre 2
D polynomials are defined and investigated. The general Hermite 2D polynomi
als are related to the two-variable Hermite polynomials but are not the sam
e. The advantage of the newly introduced Hermite and Laguerre 2D polynomial
s is that they satisfy orthogonality relations in a direct way, whereas for
the purpose of orthonormalization of the two-variable Hermite polynomials
two different sets of such polynomials are introduced which are biorthonorm
al to each other. The matrix which prays a role in the new definition of He
rmite and Laguerre 2D polynomials is in a considered sense the square root
of the matrix which plays a role in the definition of two-variable Hermite
polynomials. Two essentially different explicit representations of the Herm
ite and Laguerre 2D polynomials are derived where the first involves Jacobi
polynomials as coefficients in superpositions of special Hermits or Laguer
re 2D polynomials and the second is a superposition of products of two Herm
ite polynomials with decreasing indices and with coefficients related to th
e special Laguerre 2D polynomials. Generating functions are derived for the
Hermite and Laguerre 2D polynomials.