Formula:KLS:14.10:34

<< Formula:KLS:14.10:33

formula in Continuous q-Jacobi

Formula:KLS:14.10:35 >>

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right):=\frac{1}{\left(q;q\right)_{n}}p_{n}\!\left(x;a{\mathrm{e}^{\mathrm{i}\phi}},0,a{\mathrm{e}^{-\mathrm{i}\phi}},0\,|\,q\right)(x=\cos\left(\theta+\phi\right))}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Meixner-Pollaczek polynomial : http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : 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 14.10 of KLS.

URL links

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formula in Continuous q-Jacobi

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