Formula:KLS:14.29:02

<< Formula:KLS:14.29:01

formula in Discrete q-Hermite II

Formula:KLS:14.29:03 >>

${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)={\mathrm{i}^{-n}}q^{-\binomial{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},\mathrm{i}x}{-}{q}{-q^{n}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.29 of KLS.

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<< Formula:KLS:14.29:01

formula in Discrete q-Hermite II

Formula:KLS:14.29:03 >>