Gosper's Algorithm Research Articles

A generalization of Gosper's algorithm

We present a derivation of Gosper's algorithm which permits generalization to higher-order recurrences with constant least and most significant coefficients. Like Gosper's algorithm, the generalized algorithm requires only ‘rationalrs operations (such as gcd and resultant computations) but no factorization.

On Computing Closed Forms for Indefinite Summations

A decision procedure for finding closed forms for indefinite summation of polynomials, rational functions, quasipolynomials and quasirational functions is presented. It is also extended to deal with some non-hypergeometric sums with rational inputs, which are not summable by means of Gosper's algorithm. Discussion of its implementation, analysis of degree bounds and some illustrative examples are included.

Improvement of the Degree Setting in Gosper's Algorithm

A detailed study of the degree setting for Gosper's algorithm for indefinite hypergeometric summation is presented. In particular, we discriminate between rational and proper hypergeometric input. As a result, the critical degree bound can be improved in the former case.

The method of creative telescoping

An algorithm for definite hypergeometric summation is given. It is based, in a non-obvious way, on Gosper's algorithm for definite hypergeometric summation, and its theoretical justification relies on Bernstein's theory of holonomic systems.

A fast algorithm for proving terminating hypergeometric identities

An algorithm for proving terminating hypergeometric identities, and thus binomial coefficients identities, is presented. It is based upon Gosper's algorithm for indefinite hypergeometric summation. A MAPLE program implementing this algorithm succeeded in proving almost all known identities. Hitherto the proof of such identities was an exclusively human endeavor.