  Using Zeilberger generating function formula for the values of a discrete
analytic function in a quadrant we make connections with the theory of
structured reproducing kernel spaces, structured matrices and a generalized
moment problem. An important role is played by a Krein space realization result
of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the
origin. A key observation is that, using a simple Moebius transform, one can
reduce the study of discrete analytic functions in the upper right quadrant to
problems of function theory in the open unit disk. As an example, we associate
to each finite positive measure on $[0,2\pi]$ a discrete analytic function on
the right-upper quarter plane with values on the lattice defining a positive
definite function. Emphasis is put on the rational case, both when an
underlying Carath\'eodory function is rational and when, in the positive case,
the spectral function is rational. The rational case and the general case are
linked via the existence of a unitary dilation, possibly in a Krein space
