The hypergeometric function can be defined by

where and are vectors, , , , and if .  Our objective is to compute representations for instances of .  For example,

Various simple expressions and well known functions can be expressed in term of .  These include exponentials, binomials, logarithms, trigonometric functions, inverse trigonometric functions, incomplete Gamma function, error function, Fresnel integrals, Bessel functions, Kelvin functions, Airy functions, Struve functions, Anger J function, Weber E function, Whittaker functions, complete elliptic integrals, orthogonal polynomials, Lommel functions, polylogarithms, and Lerch function [1], [7].  For example,

hypergeometric functions are applicable to integration, differential equations, closed form summation, and difference equations [5], [6], [7].  Some methods will create answers in terms of .  An algorithm like ours can often reexpress such answers in terms of better known functions.