Difference between revisions of "Q-exponential e sub 1/q"
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The $q$-exponential $e_{\frac{1}{q}}$ is an [[entire]] function and is defined for $0 < |q| < 1$ by | The $q$-exponential $e_{\frac{1}{q}}$ is an [[entire]] function and is defined for $0 < |q| < 1$ by | ||
$$e_{\frac{1}{q}}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{ {k \choose 2} }}{(1;q)_k} z^k,$$ | $$e_{\frac{1}{q}}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{ {k \choose 2} }}{(1;q)_k} z^k,$$ | ||
| − | where ${n \choose 2}$ denotes the [[binomial coefficient]] and $(1;q)_k$ is the [[q- | + | where ${n \choose 2}$ denotes the [[binomial coefficient]] and $(1;q)_k$ is the [[q-shifted factorial]]. |
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 04:05, 21 December 2016
The $q$-exponential $e_{\frac{1}{q}}$ is an entire function and is defined for $0 < |q| < 1$ by $$e_{\frac{1}{q}}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{ {k \choose 2} }}{(1;q)_k} z^k,$$ where ${n \choose 2}$ denotes the binomial coefficient and $(1;q)_k$ is the q-shifted factorial.
