Q-Cos
From specialfunctionswiki
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
Theorem
The following formula holds: $$E_q(iz)=\mathrm{Cos}_q(z)+i\mathrm{Sin}_q(z),$$ where $E_q$ is the $q$-exponential $E_q$, $\mathrm{Cos}_q$ is the $q$-$\mathrm{Cos}$ function and $\mathrm{Sin}_q$ is the $q$-$\mathrm{Sin}$ function.
Proof
References
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: █
