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 -systems which are of the form where either , or 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 -system, and that in the case d≥2 there are -systems whose solution space is infinite dimensional. The relationship between dimensions of solution spaces in the two cases and is investigated. We prove that every consistent -system H has a non-zero solution T with . Finally we give an appropriate corollary to the Ore–Sato theorem on possible forms of solutions of -systems in this setting.