Green function for $T_{\alpha }$-Laplacian in higher dimensions

Through this article we will use a notation \begin{equation}\label{alfaLap} T_αu(x)=(1-|x|^2)Δu(x)+2 α\langle x,\nabla u(x)\rangle + (n-2-α) αu(x). \end{equation} Here, $\mathrm{\mid }x\mathrm{\mid }<1$ and $\alpha >-1$. Also, for $x=x(x_1,x_2,\dots ,x_n)\in {\mathbb{R}}^n$ we use $\mathrm{\mid }x\mathrm{\mid }=\sqrt{x_1^2+x_2^2+\dots +x_n^2},\mathrm{\nabla }=(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}),\mathrm{\Delta }=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_1^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_2^2}+\dots +\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_n^2}\mathrm{.}$ The purpose of this paper is to investigate a Dirichlet problem, corresponding to above mentioned PDE. We will specificaly consider non-homogenous boundary value problem. In that purpose the explicit formula for Green function assosiated to the operator (\ref{alfaLap}) will be calculated, and also, we will present the corresponding representation theorem.