Formula:KLS:09.07:02

<< Formula:KLS:09.07:01

formula in Meixner-Pollaczek

Formula:KLS:09.07:03 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}^{(2\phi-\pi)x}}\left|\Gamma\left(\lambda+\mathrm{i}x\right)\right|^{2}P^{(\lambda)}_{m}\!\left(x;\phi\right)P^{(\lambda)}_{n}\!\left(x;\phi\right)\,dx{}=\frac{\Gamma\left(n+2\lambda\right)}{(2\sin\phi)^{2\lambda}n!}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\lambda>0\quad 0<\phi<\pi}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.7 of KLS.

URL links

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<< Formula:KLS:09.07:01

formula in Meixner-Pollaczek

Formula:KLS:09.07:03 >>