Generalized hypergeometric function
In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. In the case of geometric series the ratio is constant. The series for the exponential function is an example, for which an/an-1 is 1/n.
In practice many interesting examples have such a property; but on the other hand the series has a non-zero radius of convergence only under restricted conditions. That means that it is usual to restrict the name to cases where there is an actual hypergeometric function that exists as an analytic function defined by such a series (and then by analytic continuation). For the standard hypergeometric series denoted by F(a, b, c; z), the convergence conditions were given by Gauss. That is the case where the ratio of coefficients is (n+a)/{(n+b)(n+c)}. Applications include to the inversion of periods of elliptic integrals.
Studies in the nineteenth century included those of Ernst Kummer, and the fundamental characterisation by Bernhard Riemann of the F-function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation (in z) for F, examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities: that effectively the entire algorithmic side of the theory was a consequence of basic facts and the use of Möbius transformations as a symmetry group.
Subsequently the hypergeometric series were generalised to several variables, for example by Appell; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. What are called q-series analogues were found. During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields.