q-Sine

The q-Analog of the Sine function, as advocated by R. W. Gosper, is defined by

$$\sin_q(z,q)={\vartheta_1(z,p)\over\vartheta_1({\textstyle{1\over 2}}\pi, p)},$$

where $\vartheta_1(z,p)$ is a Theta Function and $p$ is defined via

$$(\ln p)(\ln q)=\pi^2.$$

This is a period $2\pi$, Odd Function of unit amplitude with double and triple angle formulas and addition formulas which are analogous to ordinary Sine and Cosine. For example,

$$\sin_q(2z,q)=(q+1){\pi_q\over p_{q^2}}\cos_q(z,q^2)\sin_q(z,q^2),$$

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

See also q-Cosine, q-Factorial

References

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 Unpublished manuscript.