Formula:KLS:14.25:13

<< Formula:KLS:14.25:11

formula in Al-Salam-Carlitz II

Formula:KLS:14.25:14 >>

${\displaystyle{\displaystyle{\displaystyle w(x;a;q)V^{(a)}_{n}\!\left(x;q\right)=a^{n}(q-1)^{n}q^{-\binomial{n}{2}}\left(\mathcal{D}_{q}\right)^{n}\left[w(x;a;q)\right]}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{1}{\left(x,a^{-1}x;q\right)_{\infty}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle V^{(n)}_{\alpha}}}}$ : Al-Salam-Carlitz II polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamCarlitzII
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}^{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-derivative : http://drmf.wmflabs.org/wiki/Definition:qderiv
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.25 of KLS.

URL links

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<< Formula:KLS:14.25:11

formula in Al-Salam-Carlitz II

Formula:KLS:14.25:14 >>