Formula:KLS:09.07:06

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formula in Meixner-Pollaczek

Formula:KLS:09.07:07 >>

${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{\mathrm{i}\phi}}(\lambda-\mathrm{i}x)y(x+\mathrm{i})+2\mathrm{i}\left[x\cos\phi-(n+\lambda)\sin\phi\right]y(x){}-{\mathrm{e}^{-\mathrm{i}\phi}}(\lambda+\mathrm{i}x)y(x-\mathrm{i})=0}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle y(x)=P^{(\lambda)}_{n}\!\left(x;\phi\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1

Bibliography

Equation in Section 9.7 of KLS.

URL links

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<< Formula:KLS:09.07:05

formula in Meixner-Pollaczek

Formula:KLS:09.07:07 >>