Decision procedure for indefinite hypergeometric summation

Given a summand a n , we seek the “indefinite sum” S ( n ) determined (within an additive constant) by Formula: see text or, equivalently, by Formula: see text An algorithm is exhibited which, given a n , finds those S ( n ) with the property Formula: see text With this algorithm, we can determine, for example, the three identities Formula: see text Formula: see text and Formula: see text and we can also conclude that Formula: see text is inexpressible as S ( m ) - S (0), for any S ( n ) satisfying Eq. 2.