q-Gamma Function

A q-Analog of the Gamma Function defined by

 (1)

where is a q-Series. The -gamma function satisfies
 (2)

(Andrews 1986).

A curious identity for the functional equation
 (3)
where

 (4)

is given by
 (5)

for any .

Andrews, G. E. W. Gosper's Proof that .'' Appendix A in -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 -Trigonometric Identity.'' Amer. Math. Monthly 103, 175-177, 1996.