  Complex linear differential equations with entire coefficients are studied in
the situation where one of the coefficients is an exponential polynomial and
dominates the growth of all the other coefficients. If such an equation has an
exponential polynomial solution $f$, then the order of $f$ and of the dominant
coefficient are equal, and the two functions possess a certain duality
property. The results presented in this paper improve earlier results by some
of the present authors, and the paper adjoins with two open problems.
