Formula:KLS:09.03:19

<< Formula:KLS:09.03:17

formula in Continuous dual Hahn

Formula:KLS:09.03:20 >>

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

Substitution(s)

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

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle S_{n}}}}$ : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : 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 9.3 of KLS.

URL links

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<< Formula:KLS:09.03:17

formula in Continuous dual Hahn

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