Wilf–Zeilberger pair

In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. In particular, WZ pairs are instrumental in the evaluation of many sums involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent, and much simpler sum. Although finding WZ pairs by hand is impractical in most cases, Gosper's algorithm provides a sure method to find a function's WZ counterpart, and can be implemented in a symbolic manipulation program.

See also:
*Herbert S. Wilf
*Doron Zeilberger

Definition

Two functions, F and G, form a pair if and only if the following two conditions hold: $F(n+1,k)-F(n,k)\; =\; G(n,k+1)-G(n,k)$ and $G(n,pminfty)\; =\; 0.$Together, these conditions ensure that the sum $sum\_\{k=-infty\}^infty\; [F(n+1,k)-F(n,k)]\; =\; 0$ because the function G telescopes::

Examples

A Wilf–Zeilberger pair can be used to verify the identity $sum\_\{k=-infty\}^infty\; (-1)^k\; \{n\; choose\; k\}\; \{2k\; choose\; k\}\; 4^\{n-k\}\; =\; \{2n\; choose\; n\}$ using the proof certificate $R(n,k)=frac\{2k-1\}\{2n+1\}.$Define the following functions::2n choose n \G(n,k)&=R(n,k)F(n,k-1)end{align} Now "F" and "G" will form a Wilf-Zeilberger pair:

Further reading

Marko Petkovsek, Herbert Wilf and Doron Zeilberger, "A=B", AK Peters 1996, ISBN 1568810636. Full text online. [http://www.math.upenn.edu/~wilf/AeqB.html]

External links

* [http://www.pnas.org/cgi/reprint/75/1/40.pdf Gosper's algorithm] gives a method for generating WZ pairs when they exist.
* [http://www.math.upenn.edu/~wilf/gfology2.pdf Generatingfunctionology] provides details on the WZ method of identity certification.