Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m\_{i,\lambda}\in \mathbf{Z}$. 


We assume that the sum
$$f\_i(s)=\sum\_{\lambda\in \Lambda} \frac{m\_{i,\lambda}}{s-\lambda}$$
has only finite non zero terms.


If for $\mathrm{Re}(s)>0$, the sum $\sum\_{i=0}^\infty f\_i$ absolutely and uniformly converges to a holomorphic function $f$ defined on $\mathrm{Re}(s)>0$.


I would like to know


1. Is $\sum\_{i=1}^\infty m\_{i,\lambda}$ finite? If so, we call it $m\_\lambda$.
2. Does $f$ have a meromorphic extension to $\mathbf{C}$ with pole at $\lambda$ of residue $m\_\lambda$.
3. **Edit:** If we suppose 1, could we get 2?


Thank you!



meromorphic extension of a function
If all $m\_{i,\lambda}\geq 0$, the answers to all these questions are "yes". 
The key observation is that
$1/(s-\lambda)$ has positive real part when $s$ is in the right half-plane, while $\lambda$
is in the left half-plane. Therefore even if your series converges at ONE point $s$,
then $\Re f\_j(s)$ is a series with positive terms, from which all your statements immediately follow. Same if all $m\_{i,\lambda}\leq 0$.


EDIT. In general, without positivity assumption, this is not true. The following example was given by user fedja.


For $n\geq 2$, let $p\_n$ be rational functions with single (multiple)
pole at $-n$,
$p\_n(z)\to \phi$ uniformly in the right half-plane.
Here $\phi$ is an arbitrary bounded
analytic function
in the half-plane $\Re z>-1$.


Such functions $p\_n$
are polynomials of $1/(z+n)$ and they exist by Runge's theorem
(in fact one does not need Runge's theorem for approximating on a disk).
For each $n$, let $Q\_n$ be polynomials, partial sums of $e^z$, of sufficiently
high degree $d\_n$. The degree is such that the zeros of $Q\_n\circ p\_n$
are all in the disc $|z+n|<1/2$. This is possible because $Q\_n(z)\to e^z$
as $d\_n\to\infty$ uniformly on compact subsets of the plane,
and $e^z$ has no zeros. (Zeros of $Q\_n$ tend to $\infty$ but $p\_n$ is close to $\infty$
only near the point $-n$.)


Consider now the rational functions $g\_n=Q\_n\circ p\_n$. They converge
to the function $e^\phi$ uniformly in the right half-plane.


Zeros and poles of $g\_n$ are in the left half-plane, and they
tend to $\infty$ as $n\to\infty$. We take the set of all these zeros
and poles as our sequence $\Lambda$.


Finally, $$F\_n(z):=g\_n'(z)/g\_n(z)=
\sum\_{\lambda\in\Lambda}\frac{m\_{\lambda,n}}{z-\lambda},$$
where the sum is finite, and $m\_{\lambda,n}$ are integers. This sequence tends
to $\phi'$ uniformly in the right half-plane. But $\phi'$
was more or less arbitrary, so you can arrange it to have any singularities
in the left half-plane.
Now convert the sequence $F\_n$ into a series, if you wish.


