Formula:KLS:09.04:16

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formula in Continuous Hahn

Formula:KLS:09.04:18 >>

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

Substitution(s)

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

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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