Formula:KLS:09.04:02

<< Formula:KLS:09.04:01

formula in Continuous Hahn

Formula:KLS:09.04:03 >>

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

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
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : continuous Hahn polynomial : http://dlmf.nist.gov/18.19#P2.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.4 of KLS.

URL links

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formula in Continuous Hahn

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