Gosper's Algorithm

An algorithm for finding closed form hypergeometric identities. The algorithm treats sums whose successive terms have ratios which are rational functions. Not only does it decide conclusively whether there exists a hypergeometric sequence such that

(1)

but actually produces if it exists. If not, it produces . An outline of the algorithm follows (Petkovšek et al. 1996):

1. For the ratio which is a rational function of .

2. Write

(2)

where , , and are polynomials satisfying

(3)

for all nonnegative integers .

3. Find a nonzero polynomial solution of

(4)

if one exists.

4. Return and stop.

Petkovšek et al. (1996) describe the algorithm as "one of the landmarks in the history of computerization of the problem of closed form summation." Gosper's algorithm is vital in the operation of Zeilberger's algorithm and the machinery of Wilf-Zeilberger pairs.