q-Gamma Function

A q-analog of the gamma function defined by

(1)

where  is a q-Pochhammer symbol (Koepf 1998, p. 26; Koekoek and Swarttouw 1998). The -gamma function satisfies

(2)

where  is the gamma function (Andrews 1986).

The -gamma function is implemented in the Wolfram Language as QGamma[z, q].

The -gamma function satisfies the functional equation

(3)

with  (Koekoek and Swarttouw 1998, p. 10), which simplifies to

(4)

as . A curious identity for the functional equation

(5)

where

(6)

is given by

(7)

for any .

REFERENCES:

Andrews, G. E. "W. Gosper's Proof that ." Appendix A in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 11 and 109, 1986.

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

Koekoek, R. and Swarttouw, R. F. "The q-Gamma Function and the q-Binomial Coefficient." §0.3 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 10-11, 1998.

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

Wenchang, C. Problem 10226 and Solution. "A q-Trigonometric Identity." Amer. Math. Monthly 103, 175-177, 1996.