Formula:KLS:09.03:02

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1 2 π 0 | Γ ( a + i x ) Γ ( b + i x ) Γ ( c + i x ) Γ ( 2 i x ) | 2 S m ( x 2 ; a , b , c ) S n ( x 2 ; a , b , c ) 𝑑 x = Γ ( n + a + b ) Γ ( n + a + c ) Γ ( n + b + c ) n ! δ m , n 1 2 superscript subscript 0 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 continuous-dual-Hahn-normalized-S 𝑚 superscript 𝑥 2 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 differential-d 𝑥 Euler-Gamma 𝑛 𝑎 𝑏 Euler-Gamma 𝑛 𝑎 𝑐 Euler-Gamma 𝑛 𝑏 𝑐 𝑛 Kronecker-delta 𝑚 𝑛 {\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
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
S n subscript 𝑆 𝑛 {\displaystyle{\displaystyle{\displaystyle S_{n}}}}  : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3
δ m , n subscript 𝛿 𝑚 𝑛 {\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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