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_nxⁿ /n!; and b₀ = 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

_mF_p

when the ratio is

P(n)/Q(n)

and P has degree m, Q degree p. If

m > p + 1

we have zero as radius of convergence and so no analytic function. The classical case of Gauss therefore is

₂F₁.

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.

Thus, for example,

where (a)_n = a(a + 1)(a + 2)...(a + n - 1) is the rising factorial or Pochhammer symbol.

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 fully written therefore as

₂F₁(a,b,c;x).

The Kummer function ₁F₁(a,b;x) is known as the confluent hypergeometric function.

History and generalizations

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 (see arrangement of hyperplanes).

Hypergeometric series can be developed on Riemannian symmetric spaces and semi-simple Lie groups. Thier 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.

See also: hypergeometric function identities.

References

• (See Chapter 15.)
• (Part 1 treats hypergeometric functions on Lie groups.)