Additive Ore-Sato theorem

Let C be the field of complex numbers and C(x) be the field of rational functions in the variables x = x1, . . . , xn over C. Let Si be the shift operator with respect to xi on C(x) defined as Si(f(x1, . . . , xn)) = f(x1, . . . , xi−1, xi + 1, xi+1, . . . , xn) for any f ∈ C(x). Definition 1 (Hypergeometric and hyperarithmetic terms). A nonzero term H(x) : Nn → C is said to be hypergeometric over C(x) if there exist rational functions f1, . . . , fn ∈ C(x) such that