Closed Form

A discrete function  is called closed form (or sometimes "hypergeometric") in two variables if the ratios  and  are both rational functions. A pair of closed form functions  is said to be a Wilf-Zeilberger pair if

The term "hypergeometric function" is less commonly used to mean "closed form," and "hypergeometric series" is sometimes used to mean hypergeometric function.

A differential k-form  is said to be a closed form if .

It is worth noting that the adjective "closed" is used to describe a number of mathematical notions, e.g., the notion of closed-form solution. Loosely speaking, a solution to an equation is said to be a closed-form solution if it solves the given problem and does so in terms of functions and mathematical operations from a given generally-accepted set of "elementary notions." This particular notion of closed-form is completely separate from the notions of closedness as discussed above: In particular, the hypergeometric function (and hence, any closed-form function inheriting its properties) is considered a "special function" and is not expressible in terms of operations which are typically viewed as "elementary." What's more, certain agreed-upon truths like the insolvability of the quintic fail to be true if one extends consideration to a class of functions which includes the hypergeometric function, a result due to Klein (1877).

REFERENCES:

Klein, C. F. "Weitere Untersuchungen Über Das Ikosaeder," Mathematische Annalen 12, 503-560, 1877.

Petkovšek, 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.