q-Pi

The q-Analog of Pi $\pi_q$ can be defined by taking $a=0$ in the q-Factorial

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

giving

$$1=\sin_q({\textstyle{1\over 2}}\pi)={\pi_q\over{\mathop{\rm faq}\nolimits}^2(-{\textstyle{1\over 2}},q^2)q^{1/4}},$$

where $\sin_q(z)$ is the q-Sine. Gosper has developed an iterative algorithm for computing $\pi$ based on the algebraic Recurrence Relation

$${4\pi_{q^4}\over q^4+1}={(q^2+1)^2{\pi_q}^2\over \pi_{q^2}}-{(q^4+1){\pi_{q^2}}^2\over\pi_{q^4}}.$$