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 
.
See also
Gamma Function,
q-Beta Function,
q-Factorial,
q-Pochhammer
Symbol
Explore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
q-Gamma Function
Cite this as:
Weisstein, Eric W. "q-Gamma Function."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-GammaFunction.html
Subject classifications
