Dimensions of Solution Spaces of H-Systems∗

An H-system is a system of first-order linear homogeneous recurrence equations for a single unknown function T , with coefficients which are polynomials with complex coefficients. We consider solutions of H-systems which are of the form T : dom(T ) → C where either dom(T ) = Z, or dom(T ) = Z \ S and S is the set of integer singularities of the system. It is shown that any natural number is the dimension of the solution space of some consistent H-system, and that in the case d ≥ 2 there are H-systems whose solution space is infinite-dimensional. The relationship between dimensions of solution spaces in the two cases dom(T ) = Z and dom(T ) = Z \ S is investigated. We prove that every consistent H-system H has a non-zero solution T with dom(T ) = Z. Finally we give an appropriate corollary to the Ore-Sato theorem on possible forms of solutions of H-systems in this setting.