Completeness of systems of derivatives of Airy functions and hypercyclic operators

We study a problem of constructing new classes of hypercyclic operators on the space of all entire functions with the topology of uniform convergence on compact sets of a complex plane. This problem is closely related to the problem of completeness of some system of entire functions. It is proved that a system of successive derivatives of any non-zero solution of the Airy differential equation is complete in the space of all entire functions. This result is used to describe new classes of hypercyclic differential operators with polynomial coefficients associated with the Airy equation.