Q-exponential e sub q
From specialfunctionswiki
The $q$-exponential $e_q$ is defined by the formula $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k}.$$
Contents
Properties
Theorem: The following formula holds: $$e_q(z)=\dfrac{1}{(z;q)_{\infty}},$$ where $e_q$ is the $q$-exponential $E$ and $(z;q)_{\infty}$ denotes the q-Pochhammer symbol.
Proof: █
Theorem
The following formula holds: $$e_q(iz)=\cos_q(z)+i\sin_q(z),$$ where $e_q$ is the $q$-exponential $e_q$, $\cos_q$ is the $q$-$\cos$ function and $\sin_q$ is the $q$-$\sin$ function.
