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).

REFERENCES:

Andrews, G. E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 10, 1986.

Bailey, W. N. "Basic Hypergeometric Series." Ch. 8 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 65-72, 1935.

Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, p. 21, 1995.

Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.

Gasper, G. "Elementary Derivations of Summation and Transformation Formulas for q-Series." In Fields Inst. Comm. 14 (Ed. M. E. H. Ismail et al. ), pp. 55-70, 1997.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 107-111, 1999.

Heine, E. "Über die Reihe  ." J. reine angew. Math. 32, 210-212, 1846.

Heine, E. "Untersuchungen über die Reihe ." J. reine angew. Math. 34, 285-328, 1847.

Heine, E. Theorie der Kugelfunctionen und der verwandten Functionen, Bd. 1. Berlin: Reimer, pp. 97-125, 1878.

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 25-26, 1998.

Krattenthaler, C. "HYP and HYPQ." J. Symb. Comput. 20, 737-744, 1995.

Rogers, L. J. "On a Three-Fold Symmetry in the Elements of Heine's Series." Proc. London Math. Soc. 24, 171-179, 1893.

Slater, L. J. Generalized Hypergeometric Functions. Cambridge, England: Cambridge University Press, 1966.