Formula:KLS:09.03:19

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ω ⁢ ( x ; a , b , c ) ⁢ S n ⁡ ( x 2 ; a , b , c ) = ( δ δ ⁢ x 2 ) n ⁢ [ ω ⁢ ( x ; a + 1 2 ⁢ n ⁢ b + 1 2 ⁢ n , c + 1 2 ⁢ n ) ] 𝜔 𝑥 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 superscript 𝛿 𝛿 superscript 𝑥 2 𝑛 delimited-[] 𝜔 𝑥 𝑎 1 2 𝑛 𝑏 1 2 𝑛 𝑐 1 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c)S_{n}\!\left(x^{2};a,% b,c\right)=\left(\frac{\delta}{\delta x^{2}}\right)^{n}\left[\omega(x;a+% \textstyle\frac{1}{2}nb+\textstyle\frac{1}{2}n,c+\textstyle\frac{1}{2}n)\right% ]}}}

Substitution(s)

ω ⁢ ( x ; a , b , c ) = 1 2 ⁢ i ⁢ x ⁢ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 𝜔 𝑥 𝑎 𝑏 𝑐 1 2 imaginary-unit 𝑥 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 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c)=\frac{1}{2\mathrm{i}% x}\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}}}}


Proof

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Symbols List

S n subscript 𝑆 𝑛 {\displaystyle{\displaystyle{\displaystyle S_{n}}}}  : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1

Bibliography

Equation in Section 9.3 of KLS.

URL links

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