Covariant differential operators and Greens functions

The basic idea of this paper is to use the covariance of a partial differential operator under a suitable group action to determine suitable associated Green's functions. For instance, we offer a new proof of a formula for Green's function of the mth power $Δ^m$ of the ordinary Laplace's operator Δ in the unit disk found in a recent paper (Hayman-Korenblum, J. Anal. Math. 60 (1993), 113-133). We also study Green's functions associated with mth powers of the Poincaré invariant Laplace operator . It turns out that they can be expressed in terms of certain special functions of which the dilogarithm (m = 2) and the trilogarithm (m = 3) are the simplest instances. Finally, we establish a relationship between $Δ^m$ and : the former is up to conjugation a polynomial of the latter.