Formula:KLS:14.18:02

<< Formula:KLS:14.18:01

formula in Continuous big q-Hermite

Formula:KLS:14.18:03 >>

${\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x;a|q\right)={\mathrm{e}^{\mathrm{i}n\theta}}\,\qHyperrphis{2}{0}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}\theta}}}{-}{q}{q^{n}{\mathrm{e}^{-2\mathrm{i}\theta}}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : continuous big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.18 of KLS.

URL links

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<< Formula:KLS:14.18:01

formula in Continuous big q-Hermite

Formula:KLS:14.18:03 >>