q-Hypergeometric Function

The modern definition of the -hypergeometric function is

(1)

where is a binomial coefficient and is a q-Pochhammer symbol (Gasper and Rahman 1990; Bhatnagar 1995, p. 21; Koepf 1998, p. 25). This is the version of the -hypergeometric function implemented in the Wolfram Language as QHypergeometricPFQ[a1, ..., ar, b1, ..., bs, q, z].

An older form of definition omits the factor ,

(2)

This is the -hypergeometric function as defined by Bailey (1935), Slater (1966), Andrews (1986), and Hardy (1999).

Note that the two definitions coincide when , including the common case .

A particular case of is given by

(3)

(Andrews 1986, p. 10). A -analog of Gauss's theorem (the q-Gauss identity) due to Jacobi and Heine is given by

(4)

for (Koepf 1998, p. 40). Heine proved the transformation formula

(5)

(Andrews 1986, pp. 10-11). Rogers (1893) obtained the formulas

(6)

(7)

(Andrews 1986, pp. 10-11).

The function has the simple confluent identity

(8)

In the limit ,

(9)

where is a generalized hypergeometric function (Koepf 1998, p. 25).