q-Factorial

The q-Analog of the Factorial (by analogy with the q-Gamma Function). For $a$ an integer, the $q$-factorial is defined by

$$\mathop{\rm faq}(a,q)=1(1+q)(1+q+q^2)\cdots(1+q+\ldots+q^{a-1}).$$

A reflection formula analogous to the Gamma Function reflection formula is given by

$$\cos_q(\pi a)=\sin_q[\pi({\textstyle{1\over 2}}-a)]={\pi_q q... ...r 2}}, q^2)\mathop{\rm faq}(-(a+{\textstyle{1\over 2}}),q^2)},$$

where $\cos_q(z)$ is the q-Cosine, $\sin_q(z)$ is the q-Sine, and $\pi_q$ is q-Pi.

See also q-Beta Function, q-Cosine, q-Gamma Function, q-Pi, q-Sine

References

Gosper, R. W. ``Experiments and Discoveries in $q$-Trigonometry.'' Unpublished manuscript.