The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and less frequently studied series. The basic series is the q-analog of the ordinary hypergeometric series. There are several generalizations of the ordinary hypergeometric series, including a generalization to Riemann symmetric spaces.
where (a)n = a(a+1)(a+2)…(a+n−1) is the rising factorial, or Pochhammer symbol. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence. This series is one of 24 closely related solutions, the Kummer solutions, of the hypergeometric differential equation.
By rewriting the series as
where the nth coefficient αn is given by
it is easily seen that
It can also be shown that if c is not a negative integer, the series converges when z = 1 if ℜ(c−a−b) > 0. In this important case the value 2F1(a, b; c; 1) is given by
where Γ(z) denotes the gamma function.
The function 2F1 has several integral representations, including the Euler hypergeometric integral.
Applications of hypergeometric series include the inversion of elliptic integrals; these are constructed by taking the ratio of the two linearly independent solutions of the hypergeometric differential equation to form Schwarz-Christoffel maps of the fundamental domain to the complex projective line or Riemann sphere.
A wide range of integrals of simple functions can be expressed using the hypergeometric function, e.g.:
A limiting case of 2F1 is the Kummer function 1F1(a,b;z), known as the confluent hypergeometric function.
The series may also be written:
The series without the factor of n! in the denominator (summed over all integers n) is called the bilateral hypergeometric series.
It is currently understood that there is a very large number of such identities, and several algorithms are now known to generate and prove these identities. Some mathematicians research on the various patterns that emerge from these algorithms.
in which the ratio of successive coefficients
is a rational function of n. That is,
In practice the series is written as an exponential generating function, modifying the coefficients so that a general term of the series takes the form
and . One uses the exponential function as a 'baseline' for discussion.
Many interesting series in mathematics have the property that the ratio of successive terms is a rational function. However, when expressed as an exponential generating function, such series have a non-zero radius of convergence only under restricted conditions. Thus, by convention, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function with a non-zero radius of convergence. Such a function, and its analytic continuations, is called the hypergeometric function.
Convergence conditions were given by Carl Friedrich Gauss, who examined the case of
Here, the integers m and p refer to the degree of the polynomials P and Q, respectively, referring to the ratio
If m>p+1, the radius of convergence is zero and so there is no analytic function. 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 for F assumes that P and Q are monic and factorised, so that the notation for F includes an m-tuple that is the list of the negatives of the zeroes of P and a p-tuple of the negatives of the zeroes of Q. This is not much of a restriction: the fundamental theorem of algebra applies, and we can also absorb a leading coefficient of P or Q by redefining z. As a result of the factorisation, a general term in the series then takes the form of a ratio of products of Pochhammer symbols. Since Pochhammer notation for rising factorials is traditional it is neater to write F with the negatives of the zeros. Thus, to complete the notational example, one has
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 the 2F1, 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 Paul Emile Appell; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the q-series analogues, called the basic hypergeometric series, were given by Eduard Heine in the late nineteenth century. Here, the ratio of successive terms, instead of being a rational function of n, are considered to be a rational function of . Another generalization, the elliptic hypergeometric series, are those series where the ratio of terms is an elliptic function (a doubly periodic meromorphic function) of n.
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, Israel Gelfand and others; and applications for example to the combinatorics of arranging a number of hyperplanes in complex N-space (see arrangement of hyperplanes).
Hypergeometric series can be developed on Riemannian symmetric spaces and semi-simple Lie groups. Their importance and role can be understood through a special case: the hypergeometric series 2F1 is closely related to the Legendre polynomials, and when used in the form of spherical harmonics, it expresses, in a certain sense, the symmetry properties of the two-sphere or equivalently the rotations given by the Lie group SO(3). Concrete representations are analogous to the Clebsch-Gordan coefficients.