$\mathrm{\Delta <!--\; ??\; -->}u-<!--\; ???\; -->\frac{k}{x_n}\frac{\mathrm{\delta <!--\; ??\; -->}u}{\mathrm{\delta <!--\; ??\; -->}{x}_n}=0,$

in the upper half space ${\mathbb{R}}_{+}^{n+1}=(x_0,x_1,\dots <!--\; ???\; -->,x_n)|x_i\in <!--\; ???\; -->\mathbb{R},x_n><!--\; >\; -->0\mathrm{f}\mathrm{o}\mathrm{r}i=0,\dots <!--\; ???\; -->,n$. The operator $x_n^{\frac{2k}{n-<!--\; ???\; -->1}}(\mathrm{\Delta <!--\; ??\; -->}u-<!--\; ???\; -->\frac{k}{x_n}\frac{\mathrm{\delta <!--\; ??\; -->}u}{\mathrm{\delta <!--\; ??\; -->}{x}_n})$ is the Laplace-Beltrami operator with respect to the Riemannian metric $d{s}_k^2{x}_n^{-<!--\; ???\; -->\frac{2k}{n-<!--\; ???\; -->1}}(d{x}_0^2+d{x}_1^2+\dots <!--\; ???\; -->+d{x}_n^2$. In case k = n — 1 the Riemannian metric is the hyperbolic distance of Poincaré upper half space. The proposed functions are connected to the axially symmetric potentials studied notably by Weinstein, Huber and Leutwiler. We present the fundamental solution in case n is even using the hyperbolic metric. The main tool is the transformation of k-hyperbolic harmonic functions to eigenfunctions of the hyperbolic Laplace operator. (paper)