Formula:KLS:14.27:12

<< Formula:KLS:14.27:10

formula in Stieltjes-Wigert

Formula:KLS:14.27:13 >>

${\displaystyle{\displaystyle{\displaystyle w(x;q)S_{n}\!\left(x;q\right)=\frac{q^{n}(1-q)^{n}}{\left(q;q\right)_{n}}\left(\left(\mathcal{D}_{q}\right)^{n}w\right)(q^{n}x;q)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(-x,-qx^{-1};q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x)=\frac{\gamma}{\sqrt{\pi}}\exp\left(-\gamma^{2}{\ln^{2}}x\right),\quad x>0,\quad\textrm{with}\quad\gamma^{2}=-\frac{1}{2\ln q}=\frac{\gamma}{\sqrt{\pi}}x^{-\frac{1}{2}}\exp\left(-\gamma^{2}{\ln^{2}}x\right),x>0{\rm with}\gamma^{2}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle S_{n}}}}$ : Stieltjes-Wigert polynomial : http://drmf.wmflabs.org/wiki/Definition:StieltjesWigert
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}^{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-derivative : http://drmf.wmflabs.org/wiki/Definition:qderiv
${\displaystyle{\displaystyle{\displaystyle\mathrm{exp}}}}$ : exponential function : http://dlmf.nist.gov/4.2#E19
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2

Bibliography

Equation in Section 14.27 of KLS.

URL links

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<< Formula:KLS:14.27:10

formula in Stieltjes-Wigert

Formula:KLS:14.27:13 >>