General Hermite and Laguerre two-dimensional polynomials

Abstract

General Hermite and Laguerre two-dimensional (2D) polynomials which form a (complex) three-parameter unification of the special Hermite and Laguerre 2D polynomials are defined and investigated. The general Hermite 2D polynomials are related to the two-variable Hermite polynomials but are not the same. The advantage of the newly introduced Hermite and Laguerre 2D polynomials 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 biorthonormal to each other. The matrix which plays a role in the new definition of Hermite 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 Hermite and Laguerre 2D polynomials are derived where the first involves Jacobi polynomials as coefficients in superpositions of special Hermite or Laguerre 2D polynomials and the second is a superposition of products of two Hermite polynomials with decreasing indices and with coefficients related to the special Laguerre 2D polynomials. Generating functions are derived for the Hermite and Laguerre 2D polynomials.