Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$ I'm interested in an asymptotic expansion of the following Riemann zeta-type function
$$
\begin{align}
  \displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}, 
  \quad \Re a >-1, \, \Re b >-1, \, s>0, \tag1
\end{align} 
$$ as $s \to 0^+$.
The case $a=b$ in $(1)$ leads to the Riemann Hurwitz zeta function with the Laurent expansion near $0$:
$$
\begin{align} \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s+1}}
= \frac{1}{s}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!} \gamma_{k}(a+1)s^{k}, \quad s>0, \tag2
\end{align}
$$ where $\displaystyle \gamma_{k}(a+1)$ are the generalized Stieltjes constants with $\displaystyle \gamma_{0}(a+1)=-\Gamma'(a+1)/\Gamma(a+1)$.
What is an asymptotic expansion, as $s \to 0^+$, of $\displaystyle \zeta(s \mid a,b)$ when $a\neq b$?
 A: For $s > 0$ we have
$$
\sum_{n=1}^{\infty} \frac{1}{(n+a)^s(n+b)} - \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s+1}} = \sum_{n=1}^{\infty} \frac{a-b}{(n+a)^{s+1}(n+b)} =: g(s).
$$
The series on the right-hand side converges and is analytic on $\operatorname{Re} s > -1$, so the difference on the left-hand side can be analytically continued to this region.  Consequently, the analytic continuation $\zeta(s \mid a,b)$ of your sum satisfies
$$
\zeta(s \mid a,b) = \zeta(s+1,a+1) + g(s)
$$
for $\operatorname{Re} s > -1$, $s \neq 0$ and thus has a simple pole at $s=0$ with residue $1$.
For $|s| < 1$ we have
$$
g(s) = (a-b)\sum_{k=0}^\infty \left( \sum_{n=1}^\infty \frac{[\log(n+a)]^k}{(n+a)(n+b)} \right) \frac{(-s)^k}{k!},
$$
so for $0 < |s| < 1$
$$
\zeta(s \mid a,b) = \frac{1}{s} + \sum_{k=0}^{\infty} \left( \gamma_{k}(a+1) + (a-b)\sum_{n=1}^\infty \frac{[\log(n+a)]^k}{(n+a)(n+b)} \right) \frac{(-s)^k}{k!}.
$$
A: We may follow Euler's lead.
Euler was the first to define a constant of the form (1734)
$$
\begin{align} 
\gamma & = \lim_{N\to\infty}\left(1+\frac12+\frac13+\cdots+\frac1N-\log N\right)=0.577215\ldots. \tag1
\end{align} 
$$
Later Stieltjes found  (1885) that the Laurent series expansion around $1$ of the Riemann zeta function,
$$
\zeta(1+s) = \frac{1}{s} + \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!}\gamma_k s^k, \quad s \neq 0,\tag2
$$  is such that the scaled coefficients of the regular part of the expansion, now called the Stieltjes constants,  are given by
$$
\begin{align} 
\gamma_k& = \lim_{N\to \infty}\left(\sum_{n=1}^N \frac{\log^k n}{n}-\frac{\log^{k+1} \!N}{k+1}\right).
\end{align} \tag3
$$ 
In the same vein, J.B. Wilton (1927) and B. Berndt (1972) established that the  Laurent series expansion in the  neighbourhood of $1$ of the Hurwitz zeta function 
$$
\begin{align} \zeta(1+s,a) 
= \frac1s+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!} \gamma_{k}(a)\:s^{k}, \quad \Re a>0, \,s\neq 0, \tag4
\end{align}
$$ is such that the scaled coefficients of the regular part of the expansion, called the generalized Stieltjes constants, are given by
$$
\begin{align} 
\gamma_k(a)& = \lim_{N\to \infty}\left(\sum_{n=0}^N \frac{\log^k (n+a)}{n+a}-\frac{\log^{k+1} (N+a)}{k+1}\right), \quad \Re a>0.
\end{align} \tag5
$$ 
Do we have a form resembling the original definition of Euler's constant for our coefficients? 

Theorem. Let $a,b$ be complex numbers such that $\Re a >-1, \, \Re b >-1$. Consider the Riemann zeta type function initially defined as
  $$
\begin{align}
   \zeta(s\mid a,b) := \sum_{n=1}^{+\infty} \frac{1}{(n+a)^{s}(n+b)}, 
  \quad \Re s>0. \tag6
\end{align} 
$$
  Then the meromorphic extension of $\displaystyle \zeta(\cdot\mid a,b)$ admits the following Laurent series expansion around $0$,
  $$
\zeta(s \mid a,b) = \frac{1}{s} + \sum_{k=0}^{+\infty} \frac{(-1)^{k}}{k!}\gamma_k(a,b) s^k, \quad s \neq 0,\tag7
$$ and $$
\begin{align} 
\gamma_k(a,b)& = \lim_{N\to+\infty}\left(\sum_{n=1}^N \frac{\log^k (n+a)}{n+b}-\frac{\log^{k+1} \!N}{k+1}\right).
\end{align} \tag8
$$ 

To see this, let $a,b$ be complex numbers such that $\Re a >-1, \, \Re b >-1$.
We first assume $\Re s>0$. Observing that, for each $n \geq 1$,
$$
\left|\sum_{k=0}^{\infty}\frac{\log^k(n+a)}{n+b}\frac{(-1)^{k}}{k!}s^k\right| \leq \sum_{k=0}^{\infty}\left|\frac{\log^k(n+a)}{n+b}\right|\frac{|s|^k }{k!}<\infty
$$ and that
$$
\sum_{n=1}^{\infty}\left|\sum_{k=0}^{\infty}\frac{\log^k(n+a)}{n+b}\frac{(-1)^{k}}{k!}s^k\right|=\sum_{n=1}^{\infty}\left|\frac1{(n+a)^s(n+b)}\right| = 
\sum_{n=1}^{\infty}\frac1{|n+a|^{\Re s}|n+b|}<\infty,$$
we obtain
$$
\begin{align}
&\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\lim_{N\to+\infty}\left(\sum_{n=1}^N\frac{\log^k(n+a)}{n+b}-\frac{\log^{k+1} \!N}{k+1}\right)  s^k \\\\
&= \lim_{N\to+\infty}\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!}\left(\sum_{n=1}^N\frac{\log^k(n+a)}{n+b}-\frac{\log^{k+1} \!N}{k+1}\right) s^k \\\\
&=\lim_{N\to+\infty}\sum_{k=0}^{\infty}\left(\sum_{n=1}^N\frac{(-1)^{k}}{k!}\frac{\log^k(n+a)}{n+b}s^k -\frac{(-1)^{k}}{k!}\frac{\log^{k+1} \!N}{k+1}s^k\right) \\\\
&=\lim_{N\to+\infty}\left(\sum_{n=1}^N\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\frac{\log^k(n+a)}{n+b}s^k -\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\frac{\log^{k+1} \!N}{k+1}s^k\right) \\\\
&=\lim_{N\to+\infty}\left(\sum_{n=1}^N\frac1{(n+a)^s(n+b)} +\frac1{N^s}-\frac1s\right) \\\\
&=\zeta(s \mid a,b)-\frac1{s}.
\end{align}
$$  Then we extend the preceding identity by meromorphic continuation  to all $s \neq 0$. 
