
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. 

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