Difference between revisions of "Q-exponential e sub q"
From specialfunctionswiki
| (11 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | The $q$-exponential $e_q$ is defined by the formula | + | The $q$-exponential $e_q$ is defined for $|z|<1$ by the formula |
| − | $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k}.$$ | + | $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k},$$ |
| + | where $(q;q)_k$ denotes the [[q-shifted factorial]]. Note that this function is different than the [[q-exponential e sub 1/q |$q$-exponential $e_{\frac{1}{q}}$]]. | ||
=Properties= | =Properties= | ||
| − | + | [[Exponential e in terms of basic hypergeometric phi]] | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | [[Q-Euler formula for e sub q]] | |
| + | |||
| + | =References= | ||
| + | * {{PaperReference|q-Special functions, a tutorial|1994|Tom H. Koornwinder|prev=findme|next=findme}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 03:30, 21 December 2016
The $q$-exponential $e_q$ is defined for $|z|<1$ by the formula $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k},$$ where $(q;q)_k$ denotes the q-shifted factorial. Note that this function is different than the $q$-exponential $e_{\frac{1}{q}}$.
Properties
Exponential e in terms of basic hypergeometric phi
