Q-exponential e sub q
From specialfunctionswiki
The $q$-exponential $e_q$ is defined by the formula $$e_q(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{[k]_q!} = \displaystyle\sum_{k=0}^{\infty} \dfrac{z^k(1-q)^k}{(q;q)_k}=\displaystyle\sum_{k=0}^{\infty} z^k \dfrac{(1-q)^k}{(1-q^k)(1-q^{k-1})\ldots(1-q)},$$ where $[k]_q!$ denotes the $q$-factorial and $(q;q)_k$ denotes the $q$-Pochhammer symbol.
