Difference between revisions of "Q-exponential e sub q"
From specialfunctionswiki
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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 $ | + | 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> | ||
| + | |||
| + | {{: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.
