Formula:KLS:14.26:01

<< Al-Salam-Carlitz II

formula in Continuous q-Hermite

Formula:KLS:14.26:02 >>

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

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

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

${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHermite
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.26 of KLS.

URL links

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

<< Al-Salam-Carlitz II

formula in Continuous q-Hermite

Formula:KLS:14.26:02 >>