Formula:KLS:09.02:13

<< Formula:KLS:09.02:08

formula in Racah

Formula:KLS:09.02:14 >>

${\displaystyle{\displaystyle{\displaystyle x{\widehat{R}}_{n}\!\left(x\right)={\widehat{R}}_{n+1}\!\left(x\right)-(A_{n}+C_{n}){\widehat{R}}_{n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{R}}_{n-1}\!\left(x\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+\alpha+\beta-\gamma)(n+\alpha-\delta)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}=\left\{\begin{array}[]{ll}\displaystyle\frac{n(n+\beta)(n+\beta-\gamma-N-1)(n-\delta-N-1)}{(2n+\beta-N-1)(2n+\beta-N)}&<br/>\quad\textrm{if}\quad\alpha+1=-N\\ \displaystyle\frac{n(n+\alpha+\beta+N+1)(n+\alpha+\beta-\gamma)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}&<br/>\quad\textrm{if}\quad\beta+\delta+1=-N\\ \displaystyle\frac{n(n+\alpha+\beta+N+1)(n+\alpha-\delta)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}&<br/>\quad\textrm{if}\quad\gamma+1=-N\end{array}\right.}}}$ } &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(n+\alpha+1)(n+\alpha+\beta+1)(n+\beta+\delta+1)(n+\gamma+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}=\left\{\begin{array}[]{ll}\displaystyle\frac{(n+\beta-N)(n+\beta+\delta+1)(n+\gamma+1)(n-N)}{(2n+\beta-N)(2n+\beta-N+1)}&<br/>\quad\textrm{if}\quad\alpha+1=-N\\ \displaystyle\frac{(n+\alpha+1)(n+\alpha+\beta+1)(n+\gamma+1)(n-N)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}&<br/>\quad\textrm{if}\quad\beta+\delta+1=-N\\ \displaystyle\frac{(n+\alpha+1)(n+\alpha+\beta+1)(n+\beta+\delta+1)(n-N)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}&<br/>\quad\textrm{if}\quad\gamma+1=-N\end{array}\right.}}}$ }

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle{\widehat{R}}_{n}}}}$ : monic Racah polynomial : http://drmf.wmflabs.org/wiki/Definition:monicRacah

Bibliography

Equation in Section 9.2 of KLS.

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<< Formula:KLS:09.02:08

formula in Racah

Formula:KLS:09.02:14 >>