Formula:KLS:09.03:02

<< Formula:KLS:09.03:01

formula in Continuous dual Hahn

Formula:KLS:09.03:03 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma\left(c+\mathrm{i}x\right)}{\Gamma\left(2\mathrm{i}x\right)}\right|^{2}S_{m}\!\left(x^{2};a,b,c\right)S_{n}\!\left(x^{2};a,b,c\right)\,dx{}=\Gamma\left(n+a+b\right)\Gamma\left(n+a+c\right)\Gamma\left(n+b+c\right)n!\,\delta_{m,n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle S_{n}}}}$ : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.3 of KLS.

URL links

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formula in Continuous dual Hahn

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