Formula:KLS:09.07:12

<< Formula:KLS:09.07:10

formula in Meixner-Pollaczek

Formula:KLS:09.07:13 >>

${\displaystyle{\displaystyle{\displaystyle\omega(x;\lambda,\phi)P^{(\lambda)}_{n}\!\left(x;\phi\right)=\frac{(-1)^{n}}{n!}\left(\frac{\delta}{\delta x}\right)^{n}\left[\omega(x;\lambda+\textstyle\frac{1}{2}n,\phi)\right]}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\omega(x;\lambda,\phi)=\Gamma\left(\lambda+\mathrm{i}x\right)\Gamma\left(\lambda-\mathrm{i}x\right){\mathrm{e}^{(2\phi-\pi)x}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4

Bibliography

Equation in Section 9.7 of KLS.

URL links

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<< Formula:KLS:09.07:10

formula in Meixner-Pollaczek

Formula:KLS:09.07:13 >>