Is there any closed form expression for $E(X e^{- \mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? If not, is there any tight upper bound for this quantity? Any idea how to proceed?



Is there a closed form expression for $E(X e^{-\mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$?
Split ?


$$E(Xe^{-\mu\sqrt{X}})=E(Xe^{-\sqrt{\mu^2X}})\leq E(Xe^{-\mu^2X}\mathbf{1}\_{\mu^2X\leq 1})+E(X\mathbf{1}\_{\mu^2X\geq 1})$$


