Dear Mateusz, yes I meant the continuous part of the spectrum of the Laplacian. I knew how to write down the heat kernel when the spectrum is discrete and I naively thought that for non-compact Riemannian manifolds one should also to take into account the continuous part (I imagined a kind of of integral over the continous part). In any case, thanks a lot for the reference. Using the explicit formula in (1.7) I shall try at least to recover the Green function, that will be a first test.

The heat kernel $H(z,z',t)$ of $\mathfrak{h}$ has a continuous part since it is not compact. Using this approach do you at least recover $G_s(z,w)$ when you integrate $H$ with respect to $t$ from $0$ to $\infty$ ? In fact, because of the continuous part I'm not sure if this integral will converge...

As Noam Elkies pointed out, this summation is probably equal modulo p to the solution $u(x)$ of a linear ODE of order 2: $S:y''+a(x)y'+b(x)y=0$ for suitable functions $a(x)$ and $b(x)$. If $u(x)$ had a double root, say at $x_0$, then $u(x_0)=0$ and $u'(x_0)=0$ would be a non-trivial solution to $S$ which contradicts the uniqueness of the solution. Of course, one has to make sure that the usual proof for the uniqueness goes through in charactersitic $p$, i.e., that there is no divsion by $p$.