Formula:KLS:14.09:01

<< Al-Salam-Chihara

formula in q-Meixner-Pollaczek

Formula:KLS:14.09:02 >>

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right)=a^{-n}{\mathrm{e}^{-\mathrm{i}n\phi}}\frac{\left(a^{2};q\right)_{n}}{\left(q;q\right)_{n}}\ \qHyperrphis{3}{2}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{a^{2},0}{q}{q}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.9 of KLS.

URL links

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<< Al-Salam-Chihara

formula in q-Meixner-Pollaczek

Formula:KLS:14.09:02 >>