SumTools[Hypergeometric][MinimalZpair] - compute the minimal Z-pair
SumTools[Hypergeometric][MinimalTelescoper] - compute the minimal telescoper
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Calling Sequence
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MinimalZpair(T, n, k, En)
MinimalTelescoper(T, n, k, En)
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Parameters
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T
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hypergeometric term of n and k
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n
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name
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k
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name
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En
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name; denote the shift operator with respect to n
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Description
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L and G satisfy the following properties:
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1. is a linear recurrence operator in En with polynomial coefficients in n.
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2. is a hypergeometric term of n and k.
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3. , where denotes the shift operator with respect to k.
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4. The order of L w.r.t. En is minimal.
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The execution steps of MinimalZpair can be described as follows.
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1. Determine the applicability of Zeilberger's algorithm to .
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2. If it is proven in Step 1 that a Z-pair for does not exist, return the conclusive error message ``Zeilberger's algorithm is not applicable''. Otherwise,
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a. If is a rational function in n and k, apply the direct algorithm to compute the minimal Z-pair for .
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b. If is a nonrational term, first compute a lower bound u for the order of the telescopers for . Then compute the minimal Z-pair using Zeilberger's algorithm with u as the starting value for the guessed orders.
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Examples
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Case 1: Zeilberger's algorithm is not applicable to the input term T.
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Case 2a: Rational Function
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Case 2b: Hypergeometric
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References
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Abramov, S.A.; Geddes, K.O.; and Le, H.Q. "Computer Algebra Library for the Construction of the Minimal Telescopers." Proceedings ICMS'2002, pp. 319- 329. World Scientific, 2002.
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