Show that this probability distribution is an exponential family

Question

We define a probability distribution on the non-negative integers 0,1,2,...,as having point probabilities $$P_\eta(k)=G(k,\rho)\left(\frac{1}{\rho+\eta}\right)^{\rho}\left(\frac{\eta}{\rho+ \eta}\right)^k$$ for $\eta, \rho>0$. A explicit expression for $G(k,\rho)$ is not needed just that $\sum_{k=0}^{\infty}p_{\eta}(k)=1$. I have to show that this is an exponential family and find cumulant function. I have tried by converters and doing exponential and log tricks to get the expression: $$f(x) = \exp{\left(\frac{\theta x- b(\theta)}{a(\phi)} + c(x,\theta)\right)}$$ (where c is cumulant function), but have not succeeded. Can anyone help me?