SumTools[Hypergeometric][ZpairDirect] - perform direct algorithm to construct Zeilberger's recurrences for rational functions
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Calling Sequence
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ZpairDirect(F, n, k, En)
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Parameters
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F
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rational function 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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Let F be a rational function of n and k, En the shift operator with respect to n defined by . The ZpairDirect(F, n, k, En) command computes a Z-pair such that
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The output from ZpairDirect is a list of two elements representing the computed Z-pair provided such a pair exists.
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The main distinction between ZpairDirect and Zeilberger's algorithm is that Zeilberger's algorithm uses an item-by-item examination technique for the order of the computed difference operator L. For more information, see Zeilberger.
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The function ZpairDirect, on the other hand, uses a direct algorithm to construct a Z-pair for F. It first determines if there exists a Z-pair for F. If the answer is positive, it computes a Z-pair directly. Otherwise, it gives the conclusive error message ``there does not exist a Z-pair for F'' where F is the input rational function. When the Zeilberger routine is used, and if the input hypergeometric term T is also a rational function, ZpairDirect is invoked.
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For the ZpairDirect routine, the input F must be a rational function.
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Note: If you set infolevel[ZpairDirect] to 3, Maple prints diagnostics.
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Examples
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Set the infolevel to 3.
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ZpairDirect: "Check for the existence of a Z-pair"
ZpairDirect: "There exists a Z-pair"
ZpairDirect: "Start computing a Z-pair for the given rational function"
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If the routine cannot determine a Z-pair, Maple returns an error.
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References
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Le, H.Q. "A Direct Algorithm to Construct Zeilberger's Recurrences for Rational Functions." Proceedings FPSAC'2001, pp. 303-312. 2001.
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