How to solve the following delay differential equation?

Question

What is the solution for the following equation?

$$\frac\partial{\partial q}f(s,q)= \frac s2 f(s+2,q)$$

Note, it is known that the solution for

$$\frac\partial{\partial q}f(s,q)= s f(s+1,q)$$

will be Hurwitz Zeta $\zeta(s,-q)$.

Answer 1

This is not a solution. I just want to write down my thoughts. If they are correct then they might be useful to OP.

Let $g(s,q)=\zeta(s,-q)$ which satisfies $$\frac\partial{\partial q}g(s,q)= s g(s+1,q)\tag{1}$$

Then $$\left(\frac\partial{\partial q}\right)^2 g(s,q)= s \frac\partial{\partial q}g(s+1,q)=s (s+1)g(s+2,q)\tag{2}$$

So

$$\frac{1}{2(s+1)}\left(\frac\partial{\partial q}\right)^2 g(s,q)=\frac{s}{2}g(s+2,q)\tag{3}$$

Do Fourier transform for q we have:

$$\frac{1}{2(s+1)}(i k)^2 \bar g(s,k)=\frac{s}{2}\bar g(s+2,k)\tag{4}$$

Define $iK=\frac{1}{2(s+1)} (i k)^2$,we have: $$(i K) \bar g_1(s,K)=\frac{s}{2}\bar g_1(s+2,K)\tag{5}$$

Do inverse Fourier transform for K, we obtain:

$$\frac\partial{\partial Q}g_1(s,Q)= (s/2) g_1(s+2,Q)\tag{6}$$