Formula:KLS:09.04:03

<< Formula:KLS:09.04:02

formula in Continuous Hahn

Formula:KLS:09.04:04 >>

${\displaystyle{\displaystyle{\displaystyle(a+\iunit x)\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}=A_{n}\normctsHahnptilde{n+1}@@{x}{a}{b}{c}{d}-\left(A_{n}+C_{n}\right)\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}+C_{n}\normctsHahnptilde{n-1}@@{x}{a}{b}{c}{d}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}}}}$ : normalized continuous Hahn polynomial ${\displaystyle{\displaystyle{\displaystyle{\tilde{p}}}}}$ : http://drmf.wmflabs.org/wiki/Definition:normctsHahnptilde

Bibliography

Equation in Section 9.4 of KLS.

URL links

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formula in Continuous Hahn

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