Difference between revisions of "Q-exponential E sub q"
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| − | $$E_q(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{[k]_q! | + | If $|q|>1$ or the pair $0 < |q| <1$ and $|z| < \dfrac{1}{|1-q|}$ hold, then the $q$-exponential $E_q$ is |
| − | where $[k]_q!$ denotes the [[q-factorial|$q$-factorial]] and $( | + | $$E_q(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{[k]_q!},$$ |
| + | where $[k]_q!$ denotes the [[q-factorial|$q$-factorial]]. | ||
| + | |||
| + | =Properties= | ||
| + | [[Meromorphic continuation of q-exponential E sub q]]<br /> | ||
| + | [[Q-difference equation for q-exponential E sub q]]<br /> | ||
| + | |||
| + | =See also= | ||
| + | [[q-Cos]]<br /> | ||
| + | [[q-exponential E sub 1/q]]<br /> | ||
| + | [[q-Sin]]<br /> | ||
| + | |||
| + | =References= | ||
| + | * {{PaperReference|q-exponential and q-gamma functions. I. q-exponential functions|1994|D.S. McAnally|prev=findme|next=q-Factorial}} $(3.2)$ (calls $E_q$ $\exp_q$) | ||
| + | * {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(9.5)$ (calls $E_q(x)$ $e_q^x$) | ||
| + | * {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=findme|next=Meromorphic continuation of q-exponential E sub q}}: ($6.150$) | ||
| + | |||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 04:27, 26 December 2016
If $|q|>1$ or the pair $0 < |q| <1$ and $|z| < \dfrac{1}{|1-q|}$ hold, then the $q$-exponential $E_q$ is $$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): ($6.150$)
