Understanding a specific meromorphic function that comes from a statistical physics research problem Dear Math Stackexchange,
I'm a physics researcher working on a problem 
in quantum statistical physics. I've encountered the following
 function which I do not recognize (neither does Mathematica):
$$R(z; a,b) =\sum_{k=0}^{\infty} \frac{a^k}{1+z \, b^k},$$
where $a$ and $b$  are real parameters, $b>1$ and $0<a<b$.
In practice I need  values of this function
 for $z$ on the real positive semi-axis,
but I would like to understand it better first.
As far as I see, $R(z; a,b)$ is a meromorphic function with poles within the unit circle and
an essential singularity at $z=0$.
Can it be related to some known special function? Are there more efficient ways to compute it rather than to sum the defining series directly? I'd appreciate any advice or hint. Physical context for this problem can be found on Physics SE.  
 A: At least when $|z|>1$,
$$
R(z;a,b)=\sum_{k=0}^{+\infty}(-1)^k\frac{b^{k+1}}{b^{k+1}-a}\left(\frac1z\right)^{k+1}.
$$
Thus, $R(\,\cdot\,;a,b)$ may be expressed as a $q$-hypergeometric function ${}_2\phi_1$. In the special case when $a=1$, $R(z;1,b)=-L_1(-b/z,1/b)$ where $L_1(\,\cdot\, ;q)$ is the $q$-logarithm.
A: Completing Didier's solution, the answer for $|z|>1$ in terms of $q$-hypergeometric function is 
$$R(z; a,b) = \frac{1}{a-1} \left ( {}_2\phi_1\left [  \begin{matrix} 1/b , \, a \\   a/b \end{matrix}  ; 1/b, -1/z \right ] -1 \right ) .$$ 
A: After getting used  a bit to $q$-hypergeometric function , I found a way to re-arrange the series for  $|a|<1$ and $z>0$:
If $b>1$, $$R(z; a,b) =\frac{1}{1-a}- \sum_{k=0}^{\infty} \frac{a^k}{1+b^{-k} z^{-1}}=\frac{1}{1-a}-
  {}_2\phi_1\left [  \begin{matrix}   -1/z  , \, 1/b \\ -1/(b z) \end{matrix}  ; 1/b, a \right ] \frac{z}{1+z} .$$ 
If $b<1$, $$R(z; a,b) = {}_2\phi_1\left [  \begin{matrix}   -z  , \, b \\ -b \, z \end{matrix}  ; b, a \right ] \frac{1}{1+z} .$$ 
This covers all the relevant cases for the original physics problem.
