This is the density of a truncated Poisson: $$P(X = x \mid X > 0) = \frac{\lambda ^ x e^{- \lambda} }{x ! \left ( 1 - e^{- \lambda} \right )}$$
To show that it's member of the Exponential family I have to bring it in this form:
Exponential family:
$ f(x, \theta, \phi) = exp ( \frac{ x \theta - b(\theta}{ a(\phi)} + c(x, \phi )) $
if I do this for the Poisson distribution I get: $$ \theta = log \lambda \\ \lambda = e^\theta \\ b(\theta)= e^\theta \\ a(\phi)=1 \\ c(x,\phi)= -log(x!)$$ I struggle now with this extra term $ \frac{1}{1-e^{-\lambda}}$ from the truncated Poisson. Where does it belong to?