I have encountered an issue in a PDE (A Green's function actually). I am solving it in $(d+1)$-dimensions and I use Poincare coordinates in AdS spacetime, meaning I have a dimension $z$ and I also have $d$-dimensions $x^{\mu}, \quad \mu = 1, 2, \dots d$. The differential equation is $$(\Box - m^2)G(x,x',z,z') = z^2 \partial^2_zG - z(d-1)\partial_zG + z^2 \eta^{\mu\nu}\partial_{\mu}\partial_{\nu}G - m^2G= \frac{1}{\sqrt{g}} \delta^d(\vec{x} - \vec{x}')\delta(z-z')$$
where repeated indices are summed over, $\eta^{\mu\nu}$ is the Minkowski metric $(diag(-1,1,1, \dots))$ and $g$ is the determinant of the overall spacetime metric. The paper that I am reading suggests that a transformation of the form $$u = \frac{\eta_{\mu\nu}(x-x')^{\mu}(x-x')^{\nu}}{2zz'}$$ transforms my PDE into an ODE. This transformation is actually the chordal distance but I am encountering a few issues. The first 2 terms turn out fine and only dependent in u, but that is not the case for the 3rd term. I am not sure what I did wrong, my result is $$u \frac{\partial^2}{\partial u^2}G + (d+1)u\frac{\partial G}{\partial u} + \frac{z}{z'}\left((d-1) \frac{\partial}{\partial u} + 2u \frac{\partial^2}{\partial u^2} \right)G - m^2G = \frac{1}{\sqrt{g}} \delta^d(\vec{x} - \vec{x}')\delta(z-z').$$
The last term depends on $z$. But it should not. Also, the solution should be contain the standard hypergeometric function. For reference this is the paper that I am reading https://arxiv.org/abs/hep-th/9811257. Equation 2.7 has the exact solution.
