Difference between revisions of "Q-exponential E sub q"

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=Properties=
 
=Properties=
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[[Meromorphic continuation of q-exponential E sub q]]<br />
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<strong>Theorem:</strong> The following [[meromorphic continuation]] of $E_q$ holds:
 
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$$E_q(z)=\dfrac{1}{(z(1-q);q)_{\infty}},$$
 
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where $(z(1-q);q)_{\infty}$ denotes the [[q-Pochhammer symbol]].
 
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<strong>Proof:</strong> █
 
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=References=
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* {{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]]
 
[[Category:SpecialFunction]]

Revision as of 22:38, 16 June 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

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: █

References

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