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

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<strong>Theorem:</strong> The following formula holds:
 
<strong>Theorem:</strong> The following formula holds:
 
$$e_q(z)=\dfrac{1}{(z;q)_{\infty}},$$
 
$$e_q(z)=\dfrac{1}{(z;q)_{\infty}},$$
where $e_q$ is the [[Q-exponential e|$q$-exponential $E$]] and $(z;q)_{\infty}$ denotes the [[q-Pochhammer symbol]].
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where $e_q$ is the [[Q-exponential e|$q$-exponential $e$]] and $(z;q)_{\infty}$ denotes the [[q-Pochhammer symbol]].
 
<div class="mw-collapsible-content">
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █  
 
<strong>Proof:</strong> █  
 
</div>
 
</div>
 
</div>
 
</div>
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 +
{{:Exponential e in terms of basic hypergeometric phi}}
  
 
{{:Q-Euler formula for e sub q}}
 
{{:Q-Euler formula for e sub q}}

Revision as of 17:55, 20 May 2015

The $q$-exponential $e_q$ is defined by the formula $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k}.$$

Properties

Theorem: The following formula holds: $$e_q(z)=\dfrac{1}{(z;q)_{\infty}},$$ where $e_q$ is the $q$-exponential $e$ and $(z;q)_{\infty}$ denotes the q-Pochhammer symbol.

Proof:

Theorem

The following formula holds: $$e_q(z) = {}_1\phi_0(0;-;q;z),$$ where $e_q$ is the $q$-exponential $e$ and ${}_1\phi_0$ denotes the basic hypergeometric phi.

Proof

References

Theorem

The following formula holds: $$e_q(iz)=\cos_q(z)+i\sin_q(z),$$ where $e_q$ is the $q$-exponential $e_q$, $\cos_q$ is the $q$-$\cos$ function and $\sin_q$ is the $q$-$\sin$ function.

Proof

References