Formula:KLS:09.01:18: Difference between revisions

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Latest revision as of 08:34, 22 December 2019


δ ⁢ [ ω ⁢ ( x ; a , b , c , d ) ⁢ W n ⁡ ( x 2 ; a , b , c , d ) ] δ ⁢ x 2 = ω ⁢ ( x ; a - 1 2 , b - 1 2 ⁢ c - 1 2 , d - 1 2 ) ⁢ W n + 1 ⁡ ( x 2 ; a - 1 2 , b - 1 2 , c - 1 2 , d - 1 2 ) 𝛿 delimited-[] 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 𝛿 superscript 𝑥 2 𝜔 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 Wilson-polynomial-W 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle\frac{\delta\left[\omega(x;a,b,c,d)W% _{n}\!\left(x^{2};a,b,c,d\right)\right]}{\delta x^{2}}{}=\omega(x;a-\textstyle% \frac{1}{2},b-\textstyle\frac{1}{2}c-\textstyle\frac{1}{2},d-\textstyle\frac{1% }{2})W_{n+1}\!\left(x^{2};a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-% \textstyle\frac{1}{2},d-\textstyle\frac{1}{2}\right)}}}

Substitution(s)

ω ⁢ ( x ; a , b , c , d ) := 1 2 ⁢ i ⁢ x ⁢ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) ⁢ Γ ⁡ ( d + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 assign 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 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 𝑑 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d):=\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(d+\mathrm{i}x\right)}{% \Gamma\left(2\mathrm{i}x\right)}\right|^{2}}}}


Proof

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Symbols List

W n subscript 𝑊 𝑛 {\displaystyle{\displaystyle{\displaystyle W_{n}}}}  : Wilson polynomial : http://dlmf.nist.gov/18.25#T1.t1.r2
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 9.1 of KLS.

URL links

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