q-Gamma Function

A q-Analog of the Gamma Function defined by

$$\Gamma_q(x)\equiv {(q;q)_\infty\over (q^x;q)_\infty} (1-q)^{1-x},$$ (1)

where $(x,q)_\infty$ is a q-Series. The $q$-gamma function satisfies

$$\lim_{q\to 1^-}\Gamma_q(x)=\Gamma(x)$$ (2)

(Andrews 1986).

A curious identity for the functional equation

$f(a-b)f(a-c)f(a-d)f(a-e)-f(b)f(c)f(d)f(e)$
$=q^b f(a)f(a-b-c)f(a-b-d)f(a-b-e),\quad$	(3)

where

$$b+c+d+e=2a$$ (4)

is given by

$$f(\alpha)=\cases{ \sin(k\alpha) & for $q=1$\cr {1\over\Gamma_q(\alpha)\Gamma_q(1-\alpha)} & for $0<q<1$,\cr}$$ (5)

for any $k$.

See also q-Beta Function, q-Factorial

References

Andrews, G. E. ``W. Gosper's Proof that $\lim_{q\to 1^-}\Gamma_q(x)=\Gamma(x)$.'' 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.

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