SumTools[IndefiniteSum][Hypergeometric] - compute closed forms of indefinite sums of hypergeometric terms
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Calling Sequence
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Hypergeometric(f, k, opt)
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Parameters
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f
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hypergeometric term in k
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k
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name
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opt
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(optional) equation of the form failpoints=true or failpoints=false
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Description
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The Hypergeometric(f, k) command computes a closed form of the indefinite sum of with respect to .
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The following algorithms are used to handle indefinite sums of hypergeometric terms (see the References section):
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Koepf's extension to Gosper's algorithm, and
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the algorithm to compute additive decompositions of hypergeometric terms developed by Abramov and Petkovsek.
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The command returns if it is not able to compute a closed form.
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Examples
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Gosper's algorithm:
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The points where the telescoping equation fails:
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Koepf's extension to Gosper's algorithm:
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Abramov and Petkovsek's algorithm (note that the specified summand is not hypergeometrically summable):
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References
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Abramov, S.A., and Petkovsek, M. "Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms." Journal of Symbolic
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Gosper, R.W., Jr. "Decision Procedure for Indefinite Hypergeometric Summation." Proceedings of the National Academy of Sciences USA. Vol. 75. (1978): 40-42.
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Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.
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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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