  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.
