Q-Cos

The function $\mathrm{Cos}_q$ is defined by $$\mathrm{Cos}_q(z)=\dfrac{E_q(iz)+E_q(-iz)}{2},$$ where $E_q$ denotes the $q$-exponential $E$.

Properties

q-Euler formula for E sub q

Theorem: The following formula holds: $$D_q \mathrm{Cos}_q(az) = -a \mathrm{Sin}_q(az),$$ where $D_q$ denotes the q-difference operator, $\mathrm{Cos}$ denotes the $q$-Cosine function, and $\mathrm{Sin}$ denotes the $q$-Sine function.

Proof: █

References

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