Generalized Hypergeometric Function

The generalized hypergeometric function is given by 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 resulting generalized hypergeometric function is written

			(2)
			(3)

where is the Pochhammer symbol or rising factorial

(4)

A generalized hypergeometric function therefore has parameters of type 1 and parameters of type 2.

A number of generalized hypergeometric functions has special names. is called a confluent hypergeometric limit function, and is implemented in the Wolfram Language as Hypergeometric0F1[b, z]. (also denoted ) is called a confluent hypergeometric function of the first kind, and is implemented in the Wolfram Language as Hypergeometric1F1[a, b, z]. The function is often called "the" hypergeometric function or Gauss's hypergeometric function, and is implemented in the Wolfram Language as Hypergeometric2F1[a, b, c, x]. Arbitrary generalized hypergeometric functions are implemented as HypergeometricPFQ[a1, ...ap, b1, ..., bq, x].

The notation for generalized hypergeometric functions was introduced by Pochhammer in 1870 and modified by Barnes (1907, 1908ab; Slater 1960, p. 2; Hardy 1999, p. 111). A number of notational variations are commonly used, including

(5)

used primarily for , using square brackets instead of parentheses

(6)

(Slater 1960, p. 2), including at the end of the first row and aligning slots in the second row from the right

(7)

(Bailey 1935, p. 9), including at the end of the first row and centering each row

(8)

(Bailey 1935, p. 14), using strict matrix-like alignment of each column with columns right-aligned along their right-most elements

(9)

or

(10)

(Slater 1960, p. 31), and a variation in which rows are centered and is placed to the right separated by a vertical bar and using parenthesis

(11)

(Graham et al. 1994, p. 205) or using square brackets and a semicolon

(12)

The latter convention will be used in this work, as it provides the clearest delineation of the argument while making the most sparing use of white space in the typesetting of expressions that may contain a large number of symbolic parameters of differing lengths.

If the argument is equal to , then it is conventional to omit the argument altogether, although the trailing semicolon may be either retained or also discarded depending on notational convention. Bailey (1935, p. 9) uses the notation

(13)

although in this work, the semicolon will be omitted, i.e.,

(14)

The Kampe de Feriet function is a generalization of the generalized hypergeometric function to two variables.

The generalized hypergeometric function satisfies

(15)

for any of its numerator parameters , and

(16)

for any of its denominator parameters , where

(17)

is the differential operator (Rainville 1971, Koepf 1998, p. 27).

The generalized hypergeometric function

(18)

is a solution to the differential equation

(19)

(Bailey 1935, p. 8). The other linearly independent solution is

(20)

A generalized hypergeometric function converges absolutely on the unit circle if

(21)

(Rainville 1971, Koepf 1998).

Many sums can be written as generalized hypergeometric functions by inspection of the ratios of consecutive terms in the generating hypergeometric series. For example, for

(22)

the ratio of successive terms is

(23)

yielding

(24)

(Petkovšek et al. 1996, pp. 44-45).

Gosper (1978) discovered a slew of unusual hypergeometric function identities, many of which were subsequently proven by Gessel and Stanton (1982). An important generalization of Gosper's technique, called Zeilberger's algorithm, in turn led to the powerful machinery of the Wilf-Zeilberger pair (Zeilberger 1990).

Special hypergeometric identities include Gauss's hypergeometric theorem

(25)

for , Kummer's formula

(26)

where and is a positive integer, Saalschütz's theorem

(27)

for with a negative integer and the Pochhammer symbol, Dixon's theorem

(28)

where has a positive real part, , and , the Clausen formula

(29)

for , , , a nonpositive integer, and the Dougall-Ramanujan identity

(30)

where , , , and for , 2, ..., 6. For all these identities, is the Pochhammer symbol.

Gessel (1995) found a slew of new identities using Wilf-Zeilberger pairs, including the following:

(31)

(32)

(33)

(34)

(Petkovšek et al. 1996, pp. 135-137).

The following table gives various named identities ordered by the orders of the s they involve. Bailey (1935) gives a large number of such identities.

	Gauss's hypergeometric theorem, Kummer's theorem, Orr's theorem, Ramanujan's hypergeometric identity
	Darling's products, Dixon's theorem, Ramanujan's hypergeometric identity, Saalschütz's theorem, Thomae's theorem, Watson's theorem, Whipple's identity
	Clausen formula, Slater's formula, Whipple's transformation
	Dougall's theorem
	Whipple's identity
	Dougall-Ramanujan identity, Whipple's transformation
	Bailey's transformation

Nørlund (1955) gave the general transformation

(35)

where is the Pochhammer symbol. This identity is based on the transformation due to Euler

(36)

where is the forward difference and

(37)

(Nørlund 1955).