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).
See also
Closed-Form Solution, Differential k-form, Elementary Number, Hypergeometric Differential Equation, Hypergeometric Distribution, Hypergeometric Function, Hypergeometric Summation, Hypergeometric Term, Liouvillian Number, Rational Function, Wilf-Zeilberger Pair
Portions of this entry contributed by Christopher Stover
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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.
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Cite this as:
Stover, Christopher and Weisstein, Eric W. "Closed Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClosedForm.html