Formula:KLS:01.14:01

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formula in Some q-analogues of special functions

Formula:KLS:01.14:02 >>

${\displaystyle{\displaystyle{\displaystyle{}\mathrm{e}_{q}\!\left(z\right):=\qHyperrphis{1}{0}@@{0}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q;q\right)_{n}}=\frac{1}{\left(z;q\right)_{\infty}},\quad 0<|q|<1}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle|z|<1}}}$

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
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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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<< Transformation formulas

formula in Some q-analogues of special functions

Formula:KLS:01.14:02 >>