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Special function defined by a hypergeometric series
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). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. 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.
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.
Riemann showed that the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities.
so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics.
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
The hypergeometric function is a solution of Euler's hypergeometric differential equation
which has three regular singular points: 0,1 and ?. The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables.
Solutions at the singular points
Solutions to the hypergeometric differential equation are built out of the hypergeometric series 2F1(a,b;c;z). The equation has two linearly independent solutions. 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 the regular singular point. This gives 3 × 2 = 6 special solutions, as follows.
Around the point z = 0, two independent solutions are, if c is not a non-positive integer,
and, on condition that c is not an integer,
If c is a non-positive integer 1-m, then the first of these solutions does not exist and must be replaced by The second solution does not exist when c is an integer greater than 1, and is equal to the first solution, or its replacement, when c is any other integer. So when c is an integer, a more complicated expression must be used for a second solution, equal to the first solution multiplied by ln(z), plus another series in powers of z, involving the digamma function. See Olde Daalhuis (2010) for details.
Around z = 1, if c - a - b is not an integer, one has two independent solutions
Around z = ?, if a - b is not an integer, one has two independent solutions
Again, when the conditions of non-integrality are not met, there exist other solutions that are more complicated.
Any 3 of the above 6 solutions satisfy a linear relation as the space of solutions is 2-dimensional, giving (6 3) = 20 linear relations between them called connection formulas.
Kummer's 24 solutions
A second order Fuchsian equation with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the Coxeter groupDn of order n!2n-1. For the hypergeometric equation n=3, so the group is of order 24 and is isomorphic to the symmetric group on 4 points, and was first described by
Kummer. The isomorphism with the symmetric group is accidental and has no analogue for more than 3 singular points, and it is sometimes better to think of the group as an extension of the symmetric group on 3 points (acting as permutations of the 3 singular points) by a Klein 4-group (whose elements change the signs of the differences of the exponents at an even number of singular points). Kummer's group of 24 transformations is generated by the three transformations taking a solution F(a,b;c;z) to one of
which correspond to the transpositions (12), (23), and (34) under an isomorphism with the symmetric group on 4 points 1, 2, 3, 4. (The first and third of these are actually equal to F(a,b;c;z) whereas the second is an independent solution to the differential equation.)
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
The hypergeometric differential equation may be brought into the Q-form
by making the substitution w = uv and eliminating the first-derivative term. One finds that
Furthermore, in the case of ?=1/p, ?=1/q and ?=1/r for integers p, q, r, then the triangle tiles the sphere, the complex plane or the upper half plane according to whether ? + ? + ? - 1 is positive, zero or negative; and the s-maps are inverse functions of automorphic functions for the triangle group ?p, q, r? = ?(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):
where ?1 is the fundamental group. In other words, the monodromy is a two dimensional linear representation of the fundamental group. The monodromy group of the equation is the image of this map, i.e. the group generated by the monodromy matrices. The monodromy representation of the fundamental group can be computed explicitly in terms of the exponents at the singular points. If (?, ?'), (?, ?') and (?,?') are the exponents at 0, 1 and ?, then, taking z0 near 0, the loops around 0 and 1 have monodromy matrices
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 , see Schwarz's list or Kovacic's algorithm.
provided that z is not a real number such that it is greater than or equal to 1. and can be proved by expanding (1 - zx)-a using the binomial theorem and then integrating term by term for z with absolute value smaller than 1, and by analytic continuation elsewhere. When z is a real number greater than or equal to 1, analytic continuation must be used because (1 - zx) is zero at some point in the support of the integral, so the value of the integral may be ill-defined. This was given by Euler in 1748 and implies Euler's and Pfaff's hypergeometric transformations.
Other representations, corresponding to other branches, are given by taking the same integrand, but taking the path of integration to be a closed Pochhammer cycle enclosing the singularities in various orders. Such paths correspond to the monodromy action.
are called contiguous to 2F1(a, b; c; z). 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. This gives
relations, given by identifying any two lines on the right hand side of
where F = 2F1(a, b; c; z), F(a+) = 2F1(a + 1, b; c; z), and so on. Repeatedly applying these relations gives a linear relation over C(z) between any three functions of the form
where m, n, and l are integers.
Gauss' continued fraction
Gauss used the contiguous relations to give several ways to write a quotient of two hypergeometric functions as a continued fraction, for example:
Transformation formulas relate two hypergeometric functions at different values of the argument z.
Fractional linear transformations
Euler's transformation is
It follows by combining the two Pfaff transformations
which in turn follow from Euler's integral representation. For extension of Euler's first and second transformations, see Rathie & Paris (2007) and Rakha & Rathie (2011).
It can also be written as linear combination
If two of the numbers 1 - c, c - 1, a - b, b - a, a + b - c, c - a - b are equal or one of them is 1/2 then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first examples were given by Kummer (1836), and a complete list was given by Goursat (1881). A typical example is
Higher order transformations
If 1-c, a-b, a+b-c differ by signs or two of them are 1/3 or -1/3 then there is a cubic transformation of the hypergeometric function, connecting it to a different value of z related by a cubic equation. The first examples were given by Goursat (1881). A typical example is
There are also some transformations of degree 4 and 6. Transformations of other degrees only exist if a, b, and c are certain rational numbers (Vidunas 2005). For example,
There are many cases where hypergeometric functions can be evaluated at z = -1 by using a quadratic transformation to change z = -1 to z = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named for Ernst Kummer:
which follows from Kummer's quadratic transformations
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). Some typical examples are given by
Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. San Diego: Academic Press. ISBN0-12-336170-2. (part 1 treats hypergeometric functions on Lie groups)