Abstract
We introduce a family of solutions to a linear delay differential equation with continuous and piecewise constant arguments depending on four parameters, and we give an asymptotic expansion of any solution of this equation with respect to this family. The fundamental solutions are indexed by the zeros (counting multiplicities) of a meromorphic function in the complex plane. This function generalizes the characteristic equation which was known to characterize the oscillatory behaviour of the delay differential equation for certain values of the parameters. The proof uses the Laplace transform, Fourier series, and the adjoint equation.
