ISSN 0361-7688, Programming and Computer Software, 2006, Vol. 32, No. 2, pp. 12. Pleiades Publishing, Inc., 2006. 1 Linear homogeneous recurrence equations with Summary: ISSN 0361-7688, Programming and Computer Software, 2006, Vol. 32, No. 2, pp. 1­2. © Pleiades Publishing, Inc., 2006. 1 1 Linear homogeneous recurrence equations with polynomial coefficients and systems of such equations play a significant role in combinatorics and in the the- ory of hypergeometric functions; the question of the dimension of the space of solutions of such systems is of great importance for many problems. Let be the corresponding shift operators acting on functions (sequences) of n1, ..., nd by f(n1, ..., ni) = f(n1, ..., ni + 1, ..., nd), i = 1, ..., d. Definition 1. An H-system is a system of equations (1) where fi, gi [n1, ..., nd]\{0} and fi, gi are coprime. We say that a d-variate sequence T (i.e., a complex function defined on a subset of d) is a solution of (1) if (1) is satisfied at all those (n1, ..., ni, ..., nd) in the domain of T for which (n1, ..., ni + 1, ..., nd) belongs to the domain of T as well. We call a hypergeometric term any solution T of an H-system.