Formula:KLS:14.29:15

<< Formula:KLS:14.29:14

formula in Discrete q-Hermite II

Formula:KLS:14.29:16 >>

${\displaystyle{\displaystyle{\displaystyle w(x;q)\discrqHermiteII{n}@{x}{q}=(q-1)^{n}q^{-\binomial{n}{2}}\left(\qderiv{q}\right)^{n}\left[w(x;q)\right]}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\qPochhammer{\iunit x,-\iunit x}{q}{\infty}}=\frac{1}{\qPochhammer{-x^{2}}{q^{2}}{\infty}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}}}}$ : discrete ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite II polynomial : http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \binom{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
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 14.29 of KLS.

URL links

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<< Formula:KLS:14.29:14

formula in Discrete q-Hermite II

Formula:KLS:14.29:16 >>