Formula:KLS:09.04:23

<< Formula:KLS:09.04:22

formula in Continuous Hahn

Formula:KLS:09.04:24 >>

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

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : continuous Hahn polynomial : http://dlmf.nist.gov/18.19#P2.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\mathrm{tan}}}}$ : tangent function : http://dlmf.nist.gov/4.14#E4
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 9.4 of KLS.

URL links

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<< Formula:KLS:09.04:22

formula in Continuous Hahn

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