  Let $C^{\infty}[0,1]$ be the space of infinitely differentiable real-valued
functions on the interval [0,1]. For $f(t)\in C^{\infty}[0,1]$, we prove the
asymptotic expansion
$$f(0)=\frac12\Delta(0)-\frac14\Delta'(0)-\frac12\sum_{k=2}^\infty\frac{(-1)^k}{k!}E_k(0)\Delta^{(k)}(0),$$
where $\Delta(t)=f(t+1)+f(t)$ and $E_k(t)$ is the $k$th Euler polynomial. From
this, we obtain asymptotic expansions for the first and higher order
derivatives of the alternating Hurwitz zeta function
$$\zeta_{E}(z,q)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(n+q)^z}$$
  with respect to its first argument
$$\zeta_{E}^{(m)}(z,q)\equiv\frac{\partial^m}{\partial z^m}\zeta_E(z,q).$$
