In mathematics, a

**hypergeometric series**could in principle be any formal power series in which the ratio of successive coefficients

*a*

_{n}/

*a*

_{n-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

*a*

_{n}/

*a*

_{n-1}is 1/

*n*. In practice it is preferred to write the series as an exponential generating function, modifying the coefficients to assume the general term of the series is

*b*

_{n}

*x*

^{n}/

*n*!; and

*b*

_{0}= 1; that is, to use the exponential function as a 'baseline' for discussion.

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 elliptic integrals.

The standard notation for hypergeometric series is _{m}F_{p} when the ratio is P(*n*)/Q(*n*) and P has degree *m*, Q degree *p*. If *m* > *p*+1 we have zero radius of convergence and so no analytic function. The classical case of Gauss therefore is _{2}F_{1}. The series naturally terminates in case P(*n*) is ever 0 for *n* a natural number. If Q(n) were ever zero, the coefficients would be undefined.

The full notation assumes P and Q monic and factorised, so that F includes also an *m*-tuple of variables for the zeroes of P and a *p*-tuple for the zeroes of Q. Note that this is not much restriction: the fundamental theorem of algebra applies, and we can also absorb a leading coefficient of P or Q by redefining *x*. Since Pochhammer notation for rising factorials is traditional it is also neater to take negatives, so *a*, *b*, *c* as above rather than the zeroes which are *-a*, *-b*, *-c*. The Gauss hypergeometric function is written therefore as _{2}F_{1}(*a*,*b*,*c*;*x*).

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. There are a number of new definitions of hypergeometric series, by Aomoto, Gel'fand and others; and applications for example to the combinatorics of arranging a number of hyperplanes in complex N-space.