Formula:KLS:14.29:15

<< Formula:KLS:14.29:14

formula in Discrete q-Hermite II

Formula:KLS:14.29:16 >>

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

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(\mathrm{i}x,-\mathrm{i}x;q\right)_{\infty}}=\frac{1}{\left(-x^{2};q^{2}\right)_{\infty}}}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}}}}$ : discrete ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite II polynomial : http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 14.29 of KLS.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:KLS:14.29:14

formula in Discrete q-Hermite II

Formula:KLS:14.29:16 >>