A pair of closed form functions is said to be a Wilf-Zeilberger pair if
(1)
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The Wilf-Zeilberger formalism provides succinct proofs of known identities and allows new identities to be discovered whenever it succeeds in finding a proof certificate for a known identity. However, if the starting point is an unknown hypergeometric sum, then the Wilf-Zeilberger method cannot discover a closed form solution, while Zeilberger's algorithm can.
Wilf-Zeilberger pairs are very useful in proving hypergeometric identities of the form
(2)
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for which the addend vanishes for all outside some finite interval. Now divide by the right-hand side to obtain
(3)
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where
(4)
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Now use a rational function provided by Zeilberger's algorithm, define
(5)
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The identity (◇) then results. Summing the relation over all integers then telescopes the right side to 0, giving
(6)
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Therefore, is independent of , and so must be a constant. If is properly normalized, then it will be true that .
For example, consider the binomial coefficient identity
(7)
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the function returned by Zeilberger's algorithm is
(8)
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Therefore,
(9)
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and
(10)
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(11)
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(12)
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(13)
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Taking
(14)
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then gives the alleged identity
(15)
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Expanding and evaluating shows that the identity does actually hold, and it can also be verified that
(16)
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so (Petkovšek et al. 1996, pp. 25-27).
For any Wilf-Zeilberger pair ,
(17)
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whenever either side converges (Zeilberger 1993). In addition,
(18)
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(19)
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and
(20)
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where
(21)
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(22)
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(Amdeberhan and Zeilberger 1997). The latter identity has been used to compute Apéry's constant to a large number of decimal places (Wedeniwski).