  We investigate generalizations of the Charlier and the Meixner polynomials on
the lattice N and on the shifted lattice N+1-\beta. We combine both lattices to
obtain the bi-lattice N \cup (N+1-\beta) and show that the orthogonal
polynomials on this bi-lattice have recurrence coefficients which satisfy a
non-linear system of recurrence equations, which we can identify as a limiting
case of an (asymmetric) discrete Painlev\'e equation.
