Formula:KLS:01.14:08

<< Formula:KLS:01.14:07

formula in Some q-analogues of special functions

Formula:KLS:01.14:09 >>

${\displaystyle{\displaystyle{\displaystyle\mathrm{Sin}_{q}\!\left(z\right):=\frac{\mathrm{E}_{q}\!\left(\mathrm{i}z\right)-\mathrm{E}_{q}\!\left(-\mathrm{i}z\right)}{2\mathrm{i}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{2n+1}{2}}z^{2n+1}}{\left(q;q\right)_{2n+1}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{Sin}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-analogue of the sine function ${\displaystyle{\displaystyle{\displaystyle\mathrm{Sin}_{q}}}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qSinKLS
${\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-analogue of the exponential function ${\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}}}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qExpKLS
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 1.14 of KLS.

URL links

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<< Formula:KLS:01.14:07

formula in Some q-analogues of special functions

Formula:KLS:01.14:09 >>