Formula:KLS:14.29:14

<< Formula:KLS:14.29:13

formula in Discrete q-Hermite II

Formula:KLS:14.29:15 >>

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

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

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}^{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-derivative : http://drmf.wmflabs.org/wiki/Definition:qderiv
${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}}}}$ : discrete ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite II polynomial : http://drmf.wmflabs.org/wiki/Definition:discrqHermiteII
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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:14.29:13

formula in Discrete q-Hermite II

Formula:KLS:14.29:15 >>