Difference between revisions of "Q-exponential E sub 1/q"

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The $E_{\frac{1}{q}}$ function is defined by the formula
 
The $E_{\frac{1}{q}}$ function is defined by the formula
$$E_{\frac{1}{q}}(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{q^{ {k \choose 2} }}{[k]_q!} z^k.$$
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$$E_{\frac{1}{q}}(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{q^{\frac{k(k-1)}{2} }}{[k]_q!} z^k.$$
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[q-exponential E sub q in terms of binomial coefficient]]<br />
<strong>Theorem:</strong> The following formula holds:
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[[Q-difference equation for q-exponential E sub 1/q]]<br />
$$D_q E_{\frac{1}{q}}(az)=aE_{\frac{1}{q}}(qaz),$$
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where $D_q$ denotes the [[q-difference operator]] and $E_{\frac{1}{q}}$ denotes the [[Q-exponential E sub 1/q|$q$-exponential $E_{\frac{1}{q}}$]].
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=See Also=
<div class="mw-collapsible-content">
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[[Q-exponential E sub q]]<br />
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
=References=
 
=References=
* {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=Q-difference equation for q-exponential E sub q|next=Q-difference equation for q-exponential E sub 1/q}}: (6.153)
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* {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(9.10)$ (calls $E_{\frac{1}{q}}(x)$ $E_q^x$)
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* {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=Q-difference equation for q-exponential E sub q|next=Q-difference equation for q-exponential E sub 1/q}}: ($6.153$)
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 04:30, 26 December 2016

The $E_{\frac{1}{q}}$ function is defined by the formula $$E_{\frac{1}{q}}(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{q^{\frac{k(k-1)}{2} }}{[k]_q!} z^k.$$

Properties

q-exponential E sub q in terms of binomial coefficient
Q-difference equation for q-exponential E sub 1/q

See Also

Q-exponential E sub q

References