Wilf–Zeilberger pair

Summary

A Wilf–Zeilberger pair, or WZ pair, is a pair of hypergeometric terms $F(n, k)$ and $G(n, k)$ with $F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k)$. Summing this relation over $k$ makes the right side telescope, which proves that $\sum_k F(n, k)$ does not depend on $n$. Many binomial and hypergeometric identities can be proved this way, and the pair can be found by computer, for example with Gosper's algorithm. The rational function $G/F$ is a short certificate that anyone can check by algebra.