Oh yes I love that one. I've asked a few questions about that
https://math.stackexchange.com/questions/902514/exponential-of-a-polynomial-of-the-differential-operator
https://math.stackexchange.com/questions/1341495/exponential-of-powers-of-the-derivative-operator
https://math.stackexchange.com/questions/2281246/geometric-series-of-the-derivative-operator