The generalized hypergeometric function is given by the following series:
pFq (a1, a2, ..., ap; b1, b2, ..., bp; x) =
∞ (a1)k(a2)k ... (ap)k xk
k=0 (b1)k(b2)k ... (bq)k k!
where (a)k is the Pochhammer symbol:
(a)k = ------
Provided the series does not terminate, the generalized hypergeometric function converges for all finite x when p < q + 1. If p = q + 1, the series is convergent for all |x| < 1 and is divergent elsewhere. If p > q + 1 the series is divergent for all x except x = 0. The last case is only meaningful if the series eventually stops.
The reason for the term "hypergeometric function" for this function is its similarity to the more well-known geometric series, in that the ratio of successive terms is a rational function of the summation index k (in a geometric series, by contrast, the ratio of successive terms is a constant in the summation index). It is not hard to convert a series with rational ratios between successive terms into the form above: simply factor the numerator and denominator: the roots of the numerator become the ak's and the roots of the denominator become the bk's.
The most common special cases of the generalized hypergeometric function are 2F1(a, b; c; x), which is "the" hypergeometric function, more properly called the Gaussian hypergeometric function, and 1F1(a; b; x) (also M(a; b; x)), the confluent hypergeometric function (see that node for more details).
The Gaussian hypergeometric function, is named for the famed mathematician Karl Friederich Gauss, who had investigated the behavior of this function in the mid-19th century in conjunction with his investigations of special forms of differential equations. It is a solution to what is known as the hypergeometric differential equation:
d y dy
x(x-1) --- + ((1+a+b)x - c) -- + abc = 0
The Gaussian hypergeometric function may be arrived at by using the Frobenius method or something similar to obtain a power series solution to this equation. Because there is a theorem that states that all second-order ordinary differential equations that have up to three regular singular points can be transformed into this equation, the solutions for those equations can be seen as special cases of the hypergeometric function, many examples are given below:
ln(z+1) = z2F1(1, 1; 2; -z)
sin-1 z = z2F1(1/2, 1/2; 3/2; z2)
The complete elliptic integral of the first kind F(k|π/2) = π/22F1(1/2, 1/2; 1; k2) (this form is actually quite useless for numerical computation because it converges very slowly except when k is very small).
The Chebyshev polynomials of the first kind: Tn(x) = 2F1(-n, n; 1/2; (1-x)/2)
The Legendre polynomials Pn = 2F1(-n, n+1; 1; 1/2(1-x)).
The confluent hypergeometric function has many other functions as special cases too.
Even more functions, all of the elementary functions, and many more special functions are expressible in terms of the generalized hypergeometric function. For example:
ez = 0F0F(;;z)
cos(z) = 0F1(; 1/2; -z2/4)
A root of a quintic equation in Bring-Jerrard form:
z5 - z + α = 0
may also be expressed in terms of the generalized hypergeometric functions as:
z = α4F3(1/5, 2/5, 3/5, 4/5; 1/2, 3/4, 5/4; 3125α4/256).
Since a general quintic equation may always be reduced into Bring-Jerrard form by applying Tschirnhausen transformations, this is a simple analytical method for solving the general quintic equation.