Hypergeometric Function

A generalized hypergeometric function is a function which can be defined in the form of a hypergeometric series, i.e., a series for which the ratio of successive terms can be written

(1)

(The factor of in the denominator is present for historical reasons of notation.)

The function corresponding to , is the first hypergeometric function to be studied (and, in general, arises the most frequently in physical problems), and so is frequently known as "the" hypergeometric equation or, more explicitly, Gauss's hypergeometric function (Gauss 1812, Barnes 1908). To confuse matters even more, the term "hypergeometric function" is less commonly used to mean closed form, and "hypergeometric series" is sometimes used to mean hypergeometric function.

The hypergeometric functions are solutions to the hypergeometric differential equation, which has a regular singular point at the origin. To derive the hypergeometric function from the hypergeometric differential equation

(2)

use the Frobenius method to reduce it to

(3)

giving the indicial equation

(4)

Plugging this into the ansatz series

(5)

then gives the solution

(6)

This is the so-called regular solution, denoted

			(7)
			(8)

which converges if is not a negative integer (1) for all of and (2) on the unit circle if . Here, is a Pochhammer symbol.

The complete solution to the hypergeometric differential equation is

(9)

The hypergeometric series is convergent for arbitrary , , and for real , and for if .

Derivatives of are given by

			(10)
			(11)

(Magnus and Oberhettinger 1949, p. 8).

Hypergeometric functions with special arguments reduce to elementary functions, for example,

			(12)
			(13)
			(14)
			(15)

An integral giving the hypergeometric function is

(16)

as shown by Euler in 1748 (Bailey 1935, pp. 4-5). Barnes (1908) gave the contour integral

(17)

where and the path is curved (if necessary) to separate the poles , , ... (, 1, ...) from the poles , 1 ... (Bailey 1935, pp. 4-5; Whittaker and Watson 1990).

Curiously, at a number of very special points, the hypergeometric functions can assume rational,

			(18)
			(19)

(M. Trott, pers. comm., Aug. 5, 2002; Zucker and Joyce 2001), quadratic surd

			(20)
			(21)

(Zucker and Joyce 2001), and other exact values

			(22)
			(23)
			(24)

(Zucker and Joyce 2001, 2003).

An infinite family of rational values for well-poised hypergeometric functions with rational arguments is given by

(25)

for , 3, ..., , and

(26)

(M. L. Glasser, pers. comm., Sept. 26, 2003). This gives the particular identity

(27)

for .

A hypergeometric function can be written using Euler's hypergeometric transformations

			(28)
			(29)
			(30)
			(31)

in any one of four equivalent forms

			(32)
			(33)
			(34)

(Abramowitz and Stegun 1972, p. 559).

It can also be written as a linear combination

(35)

(Barnes 1908; Bailey 1935, pp. 3-4; Whittaker and Watson 1990, p. 291).

Kummer found all six solutions (not necessarily regular at the origin) to the hypergeometric differential equation:

			(36)
			(37)
			(38)
			(39)
			(40)
			(41)

(Abramowitz and Stegun 1972, p. 563).

Applying Euler's hypergeometric transformations to the Kummer solutions then gives all 24 possible forms which are solutions to the hypergeometric differential equation:

			(42)
			(43)
			(44)
			(45)
			(46)
			(47)
			(48)
			(49)
			(50)
			(51)
			(52)
			(53)
			(54)
			(55)
			(56)
			(57)
			(58)
			(59)
			(60)
			(61)
			(62)
			(63)
			(64)
			(65)

(Kummer 1836; Erdélyi et al. 1981, pp. 105-106).

Goursat (1881) and Erdélyi et al. (1981) give many hypergeometric transformation formulas, including several cubic transformations.

Many functions of mathematical physics can be expressed as special cases of the hypergeometric functions. For example,

(66)

where is a Legendre polynomial.

(67)

(68)

Complete elliptic integrals and the Riemann P-series can also be expressed in terms of . Special values include

	(69)
	(70)
	(71)
	(72)
	(73)
	(74)

Kummer's first formula gives

(75)

where , , , .... Many additional identities are given by Abramowitz and Stegun (1972, p. 557).

Hypergeometric functions can be generalized to generalized hypergeometric functions

(76)

A function of the form is called a confluent hypergeometric function of the first kind, and a function of the form is called a confluent hypergeometric limit function.