Almost all of the elementary functions of mathematics are either hypergeometric or ratios of hypergeometric functions. A series Σcn is hypergeometric if the ratio cn+1/cn is a rational function of n. Many of the nonelementary functions that arise in mathematics and physics also have representations as hypergeometric series.

In this chapter, we introduce three important approaches to hypergeometric functions. First, Euler's fractional integral representation leads easily to the derivation of essential identities and transformations of hypergeometric functions. A second-order linear differential equation satisfied by a hypergeometric function provides a second method. This equation was also found by Euler and then studied by Gauss. Still later, Riemann observed that a characterization of second-order equations with three regular singularities gives a powerful technique, involving minimal calculation, for obtaining formulas for hypergeometric functions. Third, Barnes expressed a hypergeometric function as a contour integral, which can be seen as a Mellin inversion formula. Some integrals that arise here are really extensions of beta integrals. They also appear in the orthogonality relations for some special orthogonal polynomials.

Perceiving their significance, Gauss gave a complete list of contiguous relations for 2F1 functions. These have numerous applications. We show how they imply some continued fraction expansions for hypergeometric functions and also contain three-term recurrence relations for hypergeometric orthogonal polynomials. We discuss one case of the latter in this chapter, namely, Jacobi polynomials.