Hypergeometric Function

The hypergeometric function is a special function defined by the hypergeometric series, a power series whose successive coefficients are related by a ratio that is itself a rational function of the summation index, a structure general enough that many elementary functions and most of the other classical special functions can be written as particular cases or limits of it. Leonhard Euler studied series of this type in the eighteenth century, but it was Carl Friedrich Gauss who gave the ordinary hypergeometric function its modern systematic treatment in an 1813 memoir, establishing its convergence properties and many of its identities, and Bernhard Riemann later characterized the same function through the differential equation it satisfies, a second order linear equation with three regular singular points. Because so many named functions in mathematical physics and combinatorics, including the Legendre and Chebyshev polynomials and the complete elliptic integrals, reduce to special cases of the hypergeometric function, it serves as a unifying framework across a large portion of classical special function theory.