Formula:KLS:09.07:16

<< Formula:KLS:09.07:15

formula in Meixner-Pollaczek

Formula:KLS:09.07:17 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{S_{n}\!\left((x-t)^{2};\lambda+\mathrm{i}t,\lambda-\mathrm{i}t,t\cot\phi\right)}{t^{n}n!}=\frac{P^{(\lambda)}_{n}\!\left(x;\phi\right)}{(\sin\phi)^{n}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle S_{n}}}}$ : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\mathrm{cot}}}}$ : cotangent function : http://dlmf.nist.gov/4.14#E7
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1

Bibliography

Equation in Section 9.7 of KLS.

URL links

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<< Formula:KLS:09.07:15

formula in Meixner-Pollaczek

Formula:KLS:09.07:17 >>