  In this paper we extend the umbral calculus, developed to deal with
difference equations on uniform lattices, to q-difference equations. We show
that many of the properties considered for shift invariant difference operators
satisfying the umbral calculus can be implemented to the case of the
q-difference operators. This q-umbral calculus can be used to provide solutions
to linear q-difference equations and q-differential delay equations. To
illustrate the method, we will apply the obtained results to the construction
of symmetry solutions for the q-heat equation and to solve a linear ordinary
second order q-difference equation
