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) |

- 1. For the ratio which is a Rational Function of .
- 2. Write

(2)

(3) - 3. Find a nonzero polynomial solution of

(4) - 4. Return and stop.

Petkovsek *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.

**References**

Gessel, I. and Stanton, D. ``Strange Evaluations of Hypergeometric Series.'' *SIAM J. Math. Anal.* **13**, 295-308, 1982.

Gosper, R. W. ``Decision Procedure for Indefinite Hypergeometric Summation.'' *Proc. Nat. Acad. Sci. USA* **75**, 40-42, 1978.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. *Concrete Mathematics: A Foundation for Computer Science, 2nd ed.*
Reading, MA: Addison-Wesley, 1994.

Lafron, J. C. ``Summation in Finite Terms.'' In *Computer Algebra Symbolic and Algebraic Computation, 2nd ed.* (Ed. B. Buchberger,
G. E. Collins, and R. Loos). New York: Springer-Verlag, 1983.

Paule, P. and Schorn, M. ``A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities.''
*J. Symb. Comput.* **20**, 673-698, 1995.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. ``Gosper's Algorithm.'' Ch. 5 in *A=B.*
Wellesley, MA: A. K. Peters, pp. 73-99, 1996.

Zeilberger, D. ``The Method of Creative Telescoping.'' *J. Symb. Comput.* **11**, 195-204, 1991.

© 1996-9

1999-05-25