Formula:KLS:01.14:02

<< Formula:KLS:01.14:01

formula in Some q-analogues of special functions

Formula:KLS:01.14:03 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-analogue of the exponential function ${\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}}}}$ used in KLS : http://drmf.wmflabs.org/wiki/Definition:qExpKLS
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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:01.14:01

formula in Some q-analogues of special functions

Formula:KLS:01.14:03 >>