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q-Cosine

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

\begin{displaymath}
\cos_q(z,q)={\vartheta_2(z,p)\over\vartheta_2(0, p)},
\end{displaymath}

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

\begin{displaymath}
(\ln p)(\ln q)=\pi^2.
\end{displaymath}

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,

\begin{displaymath}
\cos_q(2z,q)={\cos_q}^2(z,q^2)-{\sin_q}^2(z,q^2),
\end{displaymath}

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

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

See also q-Factorial, q-Sine


References

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 Unpublished manuscript.




© 1996-9 Eric W. Weisstein
1999-05-25