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 $z_n$ such that

$$t_n=z_{n+1}-z_n,$$ (1)

but actually produces $z_n$ if it exists. If not, it produces $\sum_{k=0}^{n-1} t_k$. An outline of the algorithm follows (Petkovsek 1996):

1. For the ratio $r(n)=t_{n+1}/t_n$ which is a Rational Function of $n$.

2. Write

$$r(n)={a(n)\over b(n)}{c(n+1)\over c(n)},$$ (2)

where $a(n)$, $b(n)$, and $c(n)$ are polynomials satisfying

$$\mathop{\rm GCD}\nolimits (a(n),b(n+h))=1$$ (3)

for all nonnegative integers $h$.

3. Find a nonzero polynomial solution $x(n)$ of

$$a(n)x(n+1)-b(n-1)x(n)=c(n),$$ (4)

if one exists.

4. Return $b(n-1)x(n)/c(n) t_n$ 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.

See also Hypergeometric Identity, Sister Celine's Method, Wilf-Zeilberger Pair, Zeilberger's Algorithm

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.