Formula:KLS:14.09:02

<< Formula:KLS:14.09:01

formula in q-Meixner-Pollaczek

Formula:KLS:14.09:03 >>

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right)=\frac{\left(a{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{n}}{\left(q;q\right)_{n}}{\mathrm{e}^{\mathrm{i}n(\theta+\phi)}}\ \qHyperrphis{2}{1}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}\theta}}}{a^{-1}q^{-n+1}{\mathrm{e}^{\mathrm{i}\theta}}}{q}{qa^{-1}{\mathrm{e}^{-\mathrm{i}(\theta+2\phi)}}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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

Bibliography

Equation in Section 14.9 of KLS.

URL links

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

<< Formula:KLS:14.09:01

formula in q-Meixner-Pollaczek

Formula:KLS:14.09:03 >>