“Wilf-Zeilberger Pair”的概念、定义、习题解题方法及翻译-数学辞典

单词

Wilf-Zeilberger Pair

释义

Wilf-Zeilberger Pair

A pair of Closed Form functions is said to be a Wilf-Zeilberger pair if

(1)

The Wilf-Zeilberger formalism provides succinct proofs of known identities and allows new identities to be discoveredwhenever it succeeds in finding a proof certificate for a known identity. However, if the starting point is an unknownhypergeometric sum, then the Wilf-Zeilberger method cannot discover a closed form solution, while Zeilberger'sAlgorithm can.

Wilf-Zeilberger pairs are very useful in proving Hypergeometric Identities of the form

(2)

for which the Summand

vanishes for all

outside some finite interval. Now divide by the right-hand side to obtain

(3)

where

(4)

Now use a Rational Function

provided by Zeilberger's Algorithm, define

(5)

The identity (1) then results. Summing the relation over all integers then telescopes the right side to 0, giving

(6)

Therefore,

is independent of

, and so must be a constant. If

is properly normalized, thenit will be true that

For example, consider the Binomial Coefficient identity

(7)

the function

returned by Zeilberger's Algorithm is

(8)

Therefore,

(9)

and



			(10)

Taking

(11)

then gives the alleged identity

(12)

Expanding and evaluating shows that the identity does actually hold, and it can also be verified that

(13)

(Petkovsek et al. 1996, pp. 25-27).

For any Wilf-Zeilberger pair ,

(14)

whenever either side converges (Zeilberger 1993). In addition,


	(15)

(16)

and


	(17)

where

			(18)
			(19)

(Amdeberhan and Zeilberger 1997). The latter identity has been used to compute Apéry's Constant to a large number of decimal places (Plouffe).

See also Apéry's Constant, Convergence Improvement, Zeilberger's Algorithm

References

Amdeberhan, T. and Zeilberger, D. ``Hypergeometric Series Acceleration via the WZ Method.'' Electronic J. Combinatorics 4, No. 2, R3, 1-3, 1997.http://www.combinatorics.org/Volume_4/wilftoc.html#R03. Also available at http://www.math.temple.edu/~zeilberg/mamarim/mamarimhtml/accel.html.

Cipra, B. A. ``How the Grinch Stole Mathematics.'' Science 245, 595, 1989.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. ``The WZ Phenomenon.'' Ch. 7 in A=B. Wellesley, MA: A. K. Peters, pp. 121-140, 1996.

Wilf, H. S. and Zeilberger, D. ``Rational Functions Certify Combinatorial Identities.'' J. Amer. Math. Soc. 3, 147-158, 1990.

Zeilberger, D. ``The Method of Creative Telescoping.'' J. Symb. Comput. 11, 195-204, 1991.

Zeilberger, D. ``Closed Form (Pun Intended!).'' Contemporary Math. 143, 579-607, 1993.

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