A closed form in the discrete hypergeometric sense is a function 
for which the ratios 
and 
are both rational
functions. A pair of closed form functions 
is said to be a Wilf-Zeilberger
pair if
 |
Such discrete functions are hypergeometric terms. This sense differs from an analytic 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 notion differs
from the notions of closedness above. Discrete hypergeometric
terms can be elementary, as the constant function 
shows. Analytic hypergeometric
functions are generally special functions,
but special parameters can give elementary functions.
Allowing these special functions also permits
representations of solutions of the general quintic
equation, which cannot in general be expressed by radicals (Klein 1877).
