# Formula:KLS:01.14:08

$\displaystyle {\displaystyle \qSinKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}-\qExpKLS{q}@{-\iunit z}}{2\iunit} =\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n+1}{2}}z^{2n+1}}{\qPochhammer{q}{q}{2n+1}} }$

## Proof

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## Symbols List

$\displaystyle {\displaystyle \mathrm{Sin}_{q}}$  : $\displaystyle {\displaystyle q}$ -analogue of the sine function $\displaystyle {\displaystyle \mathrm{Sin}_q}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qSinKLS
$\displaystyle {\displaystyle \mathrm{E}_{q}}$  : $\displaystyle {\displaystyle q}$ -analogue of the exponential function $\displaystyle {\displaystyle \mathrm{E}_{q}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qExpKLS
$\displaystyle {\displaystyle \mathrm{i}}$  : imaginary unit : http://dlmf.nist.gov/1.9.i
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle \binom{n}{k}}$  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1