Formula:KLS:14.04:28

<< Formula:KLS:14.04:27

formula in Continuous q-Hahn

Formula:KLS:14.05:01 >>

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b;q\right):=p_{n}\!\left(x;a{\mathrm{e}^{\mathrm{i}\phi}},b{\mathrm{e}^{\mathrm{i}\phi}},a{\mathrm{e}^{-\mathrm{i}\phi}},b{\mathrm{e}^{-\mathrm{i}\phi}}\,|\,q\right)(x=\cos\left(\theta+\phi\right))}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : little ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : Askey-Wilson polynomial : http://dlmf.nist.gov/18.28#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.4 of KLS.

URL links

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<< Formula:KLS:14.04:27

formula in Continuous q-Hahn

Formula:KLS:14.05:01 >>