We consider the complex analogues of symmetric power moments of cubic
exponential sums. These are symmetric powers of the classical Airy differential
equation. We show that their de Rham cohomologies underlie an arithmetic Hodge
structure in the sense of Anderson and we compute their Hodge numbers by means of
the irregular Hodge filtration, which is indexed by rational numbers, on their
realizations as exponential mixed Hodge structures. The main result is that all Hodge
numbers are either zero or one.
