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

Meromorphic continuation of q-exponential E sub q
Q-difference equation for q-exponential E sub q

See also

q-Cos
q-exponential E sub 1/q
q-Sin

References

• D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next) $(3.2)$ (calls $E_q$ $\exp_q$)
• 2002: Victor Kac and Pokman Cheung: Quantum Calculus ... (previous) ... (next) $(9.5)$ (calls $E_q(x)$ $e_q^x$)
• 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): ($6.150$)