Difference between revisions of "Q-exponential e sub q"
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| − | The $q$-exponential $e_q$ is defined by the formula | + | The $q$-exponential $e_q$ is defined for $0 < |q| <1$ and $|z|<1$ by the formula |
| − | $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k} | + | $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k}.$$ |
Note that this function is different than the [[q-exponential E|$q$-exponential $E$]]. | Note that this function is different than the [[q-exponential E|$q$-exponential $E$]]. | ||
Revision as of 05:06, 26 August 2015
The $q$-exponential $e_q$ is defined for $0 < |q| <1$ and $|z|<1$ by the formula $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k}.$$ Note that this function is different than the $q$-exponential $E$.
Properties
Theorem: The following formula holds: $$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.
