Difference between revisions of "Q-exponential e sub q"
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The $q$-exponential $e_q$ is defined for $0 < |q| <1$ and $|z|<1$ 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 | + | Note that this function is different than the [[q-exponential e sub 1/q |$q$-exponential $e_{\frac{1}{q}}$]]. |
=Properties= | =Properties= | ||
Revision as of 05:07, 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 $e_{\frac{1}{q}}$.
Contents
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.
