Explicit expression for a meromorphic function What is an explicit expression for the meromorphic function
$$
\sum_{n\ge 1}\frac{(-1)^n}{z+n} ?
$$

We have
$$ \sum_{n\geq 1}\left(\frac{1}{n}-\frac{1}{n+z}\right)=\gamma+\psi(1+z)\tag{1}$$
where $\psi(z)=\frac{d}{dz}\log\Gamma(z)$, by the Weierstrass product for the $\Gamma$ function. It follows that
$$ \sum_{n\geq 1}\frac{(-1)^n}{n+z}=\sum_{n\geq 1}\left(\frac{1}{2n+z}-\frac{1}{2n-1+z}\right)=\color{red}{\frac{\psi\left(\frac{1+z}{2}\right)-\psi\left(1+\frac{z}{2}\right)}{2}}.\tag{2}$$
The RHS of $(2)$ can be written in terms of $\pi$, algebraic numbers and logarithms of algebraic numbers for any $z\in\mathbb{Q}$, due to Gauss' digamma theorem. For instance, at $z=\frac{1}{2}$ the RHS of $(2)$ equals $\frac{\pi-4}{2}$ and at $z=\frac{1}{3}$ the RHS of $(2)$ equals $\frac{\pi\sqrt{3}-9+3\log 2}{3}$.
