I'm trying to determine the behavior of an asymptotic series whose $n$th term (where $n$ is even, and all odd terms vanish) is proportional to
$$\psi^{(n)}\left(\frac12+ix\right),$$
where $\psi^{(n)}$ is the $n$th polygamma function and $x>0$ is a fixed real parameter. Is there an asymptotic expression for this term at large (and even) $n$? It certainly appears to diverge, but how so? I'm hoping to find an expression like (e.g.) $x^n n!$. Simple numerical experimenting and searches for information on the polygamma function haven't helped. I did find that $\psi^{(n)}(z)=(-1)^{n+1}n!\zeta(n+1,z)$, where $\zeta$ is the Hurwitz zeta function, so the question is equivalent to asking about the asymptotic behavior of $\zeta(n,z)$ as $n\to\infty$, but I haven't found anything here either.
Note that in $$ \zeta \!\left( {n + 1,\tfrac{1}{2} + {\rm i}x} \right) = 2^{n + 1}\! \left[ {\frac{1}{{(1 + 2{\rm i}x)^{n + 1} }} + \frac{1}{{(3 + 2{\rm i}x)^{n + 1} }} + \ldots } \right] $$ the leading term dominates for large $n$. Hence $$ \psi ^{(n)} \!\left( {\tfrac{1}{2} + {\rm i}x} \right) \sim \frac{{( - 2)^{n + 1} n!}}{{(1 + 2{\rm i}x)^{n + 1} }} $$ as $n\to+\infty$, for any fixed $x\in \mathbb R$.