  We introduce a family of discrete analytic functions, called expandable
discrete analytic functions, which includes discrete analytic polynomials, and
define two products in this family. The first one is defined in a way similar
to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us
to define rational discrete analytic functions. To define the second product we
need a new space of entire functions which is contractively included in the
Fock space. We study in this space some counterparts of Schur analysis.
