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
![f(alpha)=<span style=]() {sin(kalpha) for q=1; 1/(Gamma_q(alpha)Gamma_q(1-alpha)) for 0<q<1, " src="http://mathworld.wolfram.com/images/equations/q-GammaFunction/NumberedEquation7.gif" style="height:66px; width:242px" /> |
(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.
