Hypergeometric function

It is a solution of a second-order linear ordinary differential equation (ODE).

Every second-order linear ODE with three regular singular points can be transformed into this equation.

For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953) and Olde Daalhuis (2010).

The theory of the algorithmic discovery of identities remains an active research topic.

The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum.

Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Carl Friedrich Gauss (1813).

Studies in the nineteenth century included those of Ernst Kummer (1836), and the fundamental characterisation by Bernhard Riemann (1857) of the hypergeometric function by means of the differential equation it satisfies.

The series terminates if either a or b is a nonpositive integer, in which case the function reduces to a polynomial:

In practice, most computer implementations of the hypergeometric function adopt a branch cut along the line z ≥ 1.

2F1(z) is the most common type of generalized hypergeometric series pFq, and is often designated simply F(z).

Legendre functions are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the hypergeometric function in many ways, for example

At each of the three singular points 0, 1, ∞, there are usually two special solutions of the form xs times a holomorphic function of x, where s is one of the two roots of the indicial equation and x is a local variable vanishing at a regular singular point.

A second order Fuchsian equation with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the Coxeter group W(Dn) of order 2n−1n!.

Applying Kummer's 24 = 6×4 transformations to the hypergeometric function gives the 6 = 2×3 solutions above corresponding to each of the 2 possible exponents at each of the 3 singular points, each of which appears 4 times because of the identities

by making the substitution u = wv and eliminating the first-derivative term.

The Q-form is significant in its relation to the Schwarzian derivative (Hille 1976, pp. 307–401).

Note that the connection coefficients become Möbius transformations on the triangle maps.

In the special case of λ, μ and ν real, with 0 ≤ λ,μ,ν < 1 then the s-maps are conformal maps of the upper half-plane H to triangles on the Riemann sphere, bounded by circular arcs.

Furthermore, in the case of λ=1/p, μ=1/q and ν=1/r for integers p, q, r, then the triangle tiles the sphere, the complex plane or the upper half plane according to whether λ + μ + ν – 1 is positive, zero or negative; and the s-maps are inverse functions of automorphic functions for the triangle group 〈p, q, r〉 = Δ(p, q, r).

The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in the z plane that return to the same point.

That is, when the path winds around a singularity of 2F1, the value of the solutions at the endpoint will differ from the starting point.

Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation; thus the monodromy is a mapping (group homomorphism):

In other words, the monodromy is a two dimensional linear representation of the fundamental group.

The monodromy representation of the fundamental group can be computed explicitly in terms of the exponents at the singular points.

If 1−a, c−a−b, a−b are non-integer rational numbers with denominators k,l,m then the monodromy group is finite if and only if

The Gauss hypergeometric function can be written as a John transform (Gelfand, Gindikin & Graev 2003, 2.1.2).

Gauss showed that 2F1(a, b; c; z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a, b, c, and z.

[3] Gauss used the contiguous relations to give several ways to write a quotient of two hypergeometric functions as a continued fraction, for example:

See Slater (1966, Appendix III) for a list of summation formulas at special points, most of which also appear in Bailey (1935).

There are many other formulas giving the hypergeometric function as an algebraic number at special rational values of the parameters, some of which are listed in Gessel & Stanton (1982) and Koepf (1995).