A simple construction of a fundamental solution for the extended Weinstein equation

In this article, we study the extended Weinstein equation \[ Lu=\Delta u+\frac{k}{x_n}\frac{\partial u}{\partial x_n}+\frac{\ell}{x_n^2}u, \] where \(u\) is a sufficiently smooth function defined in \(\mathbb{R}^n\) with \(x_n>0\) and \(n\ge 3\). We find a detailed construction for a fundamental solution for the operator \(L\). The fundamental solution is represented by the associated Legendre functions \(Q_\nu^\mu\).