Formula:KLS:09.01:18

<< Formula:KLS:09.01:17

formula in Wilson

Formula:KLS:09.01:20 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\delta\left[\omega(x;a,b,c,d)W_{n}\!\left(x^{2};a,b,c,d\right)\right]}{\delta x^{2}}{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})W_{n+1}\!\left(x^{2};a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d):=\frac{1}{2\mathrm{i}x}\left|\frac{\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma\left(c+\mathrm{i}x\right)\Gamma\left(d+\mathrm{i}x\right)}{\Gamma\left(2\mathrm{i}x\right)}\right|^{2}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle W_{n}}}}$ : Wilson polynomial : http://dlmf.nist.gov/18.25#T1.t1.r2
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : 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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<< Formula:KLS:09.01:17

formula in Wilson

Formula:KLS:09.01:20 >>