A q-analog of the gamma
function defined by
![Gamma_q(x)=((q;q)_infty)/((q^x;q)_infty)(1-q)^(1-x),](/images/equations/q-GammaFunction/NumberedEquation1.svg) |
(1)
|
where
is a q-Pochhammer symbol (Koepf 1998,
p. 26; Koekoek and Swarttouw 1998). The
-gamma function satisfies
![lim_(q->1^-)Gamma_q(x)=Gamma(x),](/images/equations/q-GammaFunction/NumberedEquation2.svg) |
(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
![Gamma_q(z+1)=(1-q^z)/(1-q)Gamma_q(z)](/images/equations/q-GammaFunction/NumberedEquation3.svg) |
(3)
|
with
(Koekoek and Swarttouw 1998, p. 10), which simplifies to
![Gamma(z+1)=zGamma(z)](/images/equations/q-GammaFunction/NumberedEquation4.svg) |
(4)
|
as
.
A curious identity for the functional equation
![f(a-b)f(a-c)f(a-d)f(a-e)-f(b)f(c)f(d)f(e)
=q^bf(a)f(a-b-c)f(a-b-d)f(a-b-e),](/images/equations/q-GammaFunction/NumberedEquation5.svg) |
(5)
|
where
![b+c+d+e=2a](/images/equations/q-GammaFunction/NumberedEquation6.svg) |
(6)
|
is given by
![f(alpha)={sin(kalpha) for q=1; 1/(Gamma_q(alpha)Gamma_q(1-alpha)) for 0<q<1,](/images/equations/q-GammaFunction/NumberedEquation7.svg) |
(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
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