probability - expected value of a function of two random variables

I am trying to calculate this sum (which is expected value of a function of two independent Poisson random variables):

$$\displaystyle\sum_{x=0}^{\infty}\sum_{y=0}^{\infty}\frac{1}{a^x+a^y-a^{x+y}}\frac{e^{-\lambda}(\lambda)^x}{x!}\frac{e^{-\lambda}(\lambda)^y}{y!}$$

Is there any closed form solution for this?

Answer

I calculated this in Mathematica, but it couldn't find a closed form for this sum, nor for the integral over $x$ and $y$ from $0$ to $\infty$, so I guess a nice closed form doesn't exist.