  Differential difference algebras were introduced by Mansfield and Szanto,
which arose naturally from differential difference equations. In this paper, we
investigate the Gelfand-Kirillov dimension of differential difference algebras.
We give a lower bound of the Gelfand-Kirillov dimension of a differential
difference algebra and a sufficient condition under which the lower bound is
reached; we also find an upper bound of this Gelfand-Kirillov dimension under
some specific conditions and construct an example to show that this upper bound
can not be sharpened any more.
