ORTHOGONAL POLYNOMIALS ASSOCIATED WITH THE DIRAC OPERATOR IN EUCLIDEAN-SPACE

The author considers the possibility of generalizing the theory of classical polynomials to the higher dimensional case. The starting point is the splitting up of the second order differential operator of these polynomials into the derivation operator, considered as an operator between Hilbert spaces and its adjoint. In the case of several dimensions the derivation operator is replaced by the Dirac operator. As however the set of polynomials in the vector variable x is not dense in the Hilbert modules considered, first a decomposition of these modules in terms of spherical monogenic functions is proved. Then by applying the theory to each of the constituents, generalizations of the Gegenbauer and the Hermite polynomials are obtained.