q-Cosine

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

$$\cos_q(z,q)={\vartheta_2(z,p)\over\vartheta_2(0, p)},$$

where $\vartheta_2(z,p)$ is a Theta Function and $p$ is defined via

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

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

$$\cos_q(2z,q)={\cos_q}^2(z,q^2)-{\sin_q}^2(z,q^2),$$

where $\sin_q(z,a)$ is the q-Sine, and $\pi_q$ is q-Pi. The $q$-cosine also satisfies

$$\cos_q(\pi a)={\sum_{n=-\infty}^\infty (-1)^nq^{(n+a)^2}\over \sum_{n=-\infty}^\infty (-1)^n q^{n^2}}.$$

See also q-Factorial, q-Sine

References

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 Unpublished manuscript.