Closed Form

A discrete Function $A(n,k)$ is called closed form (or sometimes ``hypergeometric'') in two variables if the ratios $A(n+1,k)/A(n,k)$ and $A(n,k+1)/A(n,k)$ are both Rational Functions. A pair of closed form functions $(F,G)$ is said to be a Wilf-Zeilberger Pair if

$$F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k).$$

See also Rational Function, Wilf-Zeilberger Pair

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, p. 141, 1996.

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