q-Gamma Function

A q-analog of the gamma function defined by


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


where Gamma(z) is the gamma function (Andrews 1986).

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

The q-gamma function satisfies the functional equation


with Gamma_q(1)=1 (Koekoek and Swarttouw 1998, p. 10), which simplifies to


as q->1^-. A curious identity for the functional equation




is given by

 f(alpha)={sin(kalpha)   for q=1; 1/(Gamma_q(alpha)Gamma_q(1-alpha))   for 0<q<1,

for any k.

See also

Gamma Function, q-Beta Function, q-Factorial, q-Pochhammer Symbol

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Andrews, G. E. "W. Gosper's Proof that lim_(q->1^-)Gamma_q(x)=Gamma(x)." 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.

Referenced on Wolfram|Alpha

q-Gamma Function

Cite this as:

Weisstein, Eric W. "q-Gamma Function." From MathWorld--A Wolfram Web Resource.

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