Q-exponential E sub q

If $|q|>1$ or the pair $0 < |q| <1$ and $|z| < \dfrac{1}{|1-q|}$ hold, then the $q$-exponential $E_q$ is $$E_q(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{[k]_q!},$$ where $[k]_q!$ denotes the $q$-factorial.

Properties

Theorem: The following meromorphic continuation of $E_q$ holds: $$E_q(z)=\dfrac{1}{(z(1-q);q)_{\infty}},$$ where $(z(1-q);q)_{\infty}$ denotes the q-Pochhammer symbol.

Proof: █

Theorem: The following formula holds: $$D_q E_q(z) = aE_q(az),$$ where $D_q$ is the $q$-difference operator and $E_q$ is the $q$-exponential $E_q$.

Proof: █