I have the joint PMF

$\exp(y_1 \ln(\lambda)+y_2 \ln(c)+y_2\ln(\lambda)-\ln(y_1!y_2!)-\lambda(1+c))$

for a constant $c>0$. In canonical representation and mixed parameterization I have $\mathbf{\theta}=(\ln(c),\ln(\lambda))$ and $\mathbf{t}=(v,u)^T$ with $v=Y_2$ and $u=Y_1+Y_2$.

What is the marginal PMF $f(v)$?