Density of polyanalytic polynomials in complex and quaternionic polyanalytic weighted Bergman spaces

We introduce the concepts of complex polyanalytic weighted Bergman spaces and of quaternionic polyanalytic weighted Bergman spaces of first and sec-ond kind. In these spaces we then prove qualitative and quantitative results in approximation by polyanalytic polynomials. The quantitative approxima-tion results are given in terms of higher order Lp-moduli of smoothness and in terms of the best approximation quantity.