Formula:KLS:14.09:12

<< Formula:KLS:14.09:09

formula in q-Meixner-Pollaczek

Formula:KLS:14.09:13 >>

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

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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 14.9 of KLS.

URL links

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<< Formula:KLS:14.09:09

formula in q-Meixner-Pollaczek

Formula:KLS:14.09:13 >>