Formula:KLS:09.04:18

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formula in Continuous Hahn

Formula:KLS:09.04:19 >>

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

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d)=\Gamma\left(a+\iunit x\right)\Gamma\left(b+\iunit x\right)\Gamma\left(c-\iunit x\right)\Gamma\left(d-\iunit x\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : continuous Hahn polynomial : http://dlmf.nist.gov/18.19#P2.p1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.4 of KLS.

URL links

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formula in Continuous Hahn

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