Infinite Order Differential Operators in Spaces of Entire Functions

We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane. We obtain integral representations of some particular cases of these operators and apply these results to obtain explicit solutions to some Cauchy problems for diffusion equations with nonconstant drift term.