Formula:KLS:09.03:03

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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 ⁡ ( a + b ) ⁢ Γ ⁡ ( a + c ) ⁢ Γ ⁡ ( b - a ) ⁢ Γ ⁡ ( c - a ) Γ ⁡ ( - 2 ⁢ a ) ⁢ ∑ k = 0 , 1 , 2 ⁢ … a + k < 0 ( 2 ⁢ a ) k ⁢ ( a + 1 ) k ⁢ ( a + b ) k ⁢ ( a + c ) k ( a ) k ⁢ ( a - b + 1 ) k ⁢ ( a - c + 1 ) k ⁢ k ! ⁢ ( - 1 ) k ⁢ S m ⁡ ( - ( a + k ) 2 ; a , b , c ) ⁢ S n ⁡ ( - ( a + k ) 2 ; a , b , c ) = 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 𝑎 𝑏 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑏 𝑎 Euler-Gamma 𝑐 𝑎 Euler-Gamma 2 𝑎 subscript 𝑘 0 1 2 … 𝑎 𝑘 0 Pochhammer-symbol 2 𝑎 𝑘 Pochhammer-symbol 𝑎 1 𝑘 Pochhammer-symbol 𝑎 𝑏 𝑘 Pochhammer-symbol 𝑎 𝑐 𝑘 Pochhammer-symbol 𝑎 𝑘 Pochhammer-symbol 𝑎 𝑏 1 𝑘 Pochhammer-symbol 𝑎 𝑐 1 𝑘 𝑘 superscript 1 𝑘 continuous-dual-Hahn-normalized-S 𝑚 superscript 𝑎 𝑘 2 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑎 𝑘 2 𝑎 𝑏 𝑐 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{}+\frac{\Gamma% \left(a+b\right)\Gamma\left(a+c\right)\Gamma\left(b-a\right)\Gamma\left(c-a% \right)}{\Gamma\left(-2a\right)}{}\sum_{\begin{array}[]{c}{\scriptstyle k=0,1,% 2\ldots}\\ {\scriptstyle a+k<0}\end{array}}\frac{{\left(2a\right)_{k}}{\left(a+1\right)_{% k}}{\left(a+b\right)_{k}}{\left(a+c\right)_{k}}}{{\left(a\right)_{k}}{\left(a-% b+1\right)_{k}}{\left(a-c+1\right)_{k}}k!}(-1)^{k}{}S_{m}\!\left(-(a+k)^{2};a,% b,c\right)S_{n}\!\left(-(a+k)^{2};a,b,c\right){}=\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
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
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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