compute closed forms of indefinite sums of hypergeometric terms
Hypergeometric(f, k, opt)
hypergeometric term in k
(optional) equation of the form failpoints=true or failpoints=false
The Hypergeometric(f, k) command computes a closed form of the indefinite sum of f with respect to k.
The following algorithms are used to handle indefinite sums of hypergeometric terms (see the References section):
Koepf's extension to Gosper's algorithm, and
the algorithm to compute additive decompositions of hypergeometric terms developed by Abramov and Petkovsek.
If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair s,p,q, where s is the closed form of the indefinite sum of f w.r.t. k, as above, and p,q are lists of points where f does not exist or the computed sum s is undefined or improper, respectively (see SumTools[IndefiniteSum][Indefinite] for more detailed help).
The command returns FAIL if it is not able to compute a closed form.
f ≔ 4⁢n−1⁢binomial⁡2⁢n,n22⁢n−12⁢42⁢n
The points where the telescoping equation fails:
f ≔ binomial⁡2⁢n−3,n4n
s,fp ≔ Hypergeometric⁡f,n,'failpoints'
Error, numeric exception: division by zero
Koepf's extension to Gosper's algorithm:
f ≔ binomial⁡m,j2⁢binomial⁡m,k2⁢binomial⁡2⁢m+n3−j−k,2⁢m
Abramov and Petkovsek's algorithm (note that the specified summand is not hypergeometrically summable):
f ≔ n2−2⁢n−1⁢2nn+1⁢n2⁢n+3!
Error, (in SumTools:-Hypergeometric:-Gosper) no solution found
Abramov, S.A., and Petkovsek, M. "Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms." Journal of Symbolic Computing. Vol. 33. (2002): 521-543.
Gosper, R.W., Jr. "Decision Procedure for Indefinite Hypergeometric Summation." Proceedings of the National Academy of Sciences USA. Vol. 75. (1978): 40-42.
Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.
Abramov, S.A. and Petkovsek, M. "Gosper's Algorithm, Accurate Summation, and the discrete Newton-Leibniz formula." Proceedings ISSAC'05. (2005): 5-12.
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