I recently encountered a particular delay PDE in my work, the solution of which corresponds to the Laplace transform of some probability distribution. I'm having trouble to solve this equation. The solution $f(q,s)$ defined on $q,s\geq 0$ should verify $f(0,s) = 1 \forall s$ and $f(q, s\to \infty) = \delta_{q,0}$ (the Kronecker delta). It should also the verify the following PDE 
$$\partial_q f(q,s-1) = -\sqrt{s(2+s)}f(q,s) $$
It looks a bit like a "deformed" Hurwitz zeta function relation (which is $\partial_q \zeta(q,s) = -s\zeta(q,s+1)$). I looked at some simple ansätze but couldn't come up with anything useful. I also am not sure if the boundary conditions I specified above are enough to obtain a unique solution...

I would be glad if you have any suggestions.
Thanks a lot !