Formula:KLS:09.02:07

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λ ⁢ ( x ) ⁢ R n ⁡ ( λ ⁢ ( x ) ) = A n ⁢ R n + 1 ⁡ ( λ ⁢ ( x ) ) - ( A n + C n ) ⁢ R n ⁡ ( λ ⁢ ( x ) ) + C n ⁢ R n - 1 ⁡ ( λ ⁢ ( x ) ) 𝜆 𝑥 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 subscript 𝐴 𝑛 Racah-polynomial-R 𝑛 1 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 subscript 𝐶 𝑛 Racah-polynomial-R 𝑛 1 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 {\displaystyle{\displaystyle{\displaystyle\lambda(x)R_{n}\!\left(\lambda(x)% \right)=A_{n}R_{n+1}\!\left(\lambda(x)\right)-\left(A_{n}+C_{n}\right)R_{n}\!% \left(\lambda(x)\right)+C_{n}R_{n-1}\!\left(\lambda(x)\right)}}}

Substitution(s)

C n = n ⁢ ( n + α + β - γ ) ⁢ ( n + α - δ ) ⁢ ( n + β ) ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 1 ) = { n ⁢ ( n + β ) ⁢ ( n + β - γ - N - 1 ) ⁢ ( n - δ - N - 1 ) ( 2 ⁢ n + β - N - 1 ) ⁢ ( 2 ⁢ n + β - N ) < b r / >   if   α + 1 = - N n ⁢ ( n + α + β + N + 1 ) ⁢ ( n + α + β - γ ) ⁢ ( n + β ) ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 1 ) < b r / >   if   β + δ + 1 = - N n ⁢ ( n + α + β + N + 1 ) ⁢ ( n + α - δ ) ⁢ ( n + β ) ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 1 ) < b r / >   if   γ + 1 = - N subscript 𝐶 𝑛 𝑛 𝑛 𝛼 𝛽 𝛾 𝑛 𝛼 𝛿 𝑛 𝛽 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 cases 𝑛 𝑛 𝛽 𝑛 𝛽 𝛾 𝑁 1 𝑛 𝛿 𝑁 1 2 𝑛 𝛽 𝑁 1 2 𝑛 𝛽 𝑁 fragments b r italic-  if italic-  α 1 N 𝑛 𝑛 𝛼 𝛽 𝑁 1 𝑛 𝛼 𝛽 𝛾 𝑛 𝛽 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 fragments b r italic-  if italic-  β δ 1 N 𝑛 𝑛 𝛼 𝛽 𝑁 1 𝑛 𝛼 𝛿 𝑛 𝛽 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 fragments b r italic-  if italic-  γ 1 N {\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.}}} } &

A n = ( n + α + 1 ) ⁢ ( n + α + β + 1 ) ⁢ ( n + β + δ + 1 ) ⁢ ( n + γ + 1 ) ( 2 ⁢ n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β + 2 ) = { ( n + β - N ) ⁢ ( n + β + δ + 1 ) ⁢ ( n + γ + 1 ) ⁢ ( n - N ) ( 2 ⁢ n + β - N ) ⁢ ( 2 ⁢ n + β - N + 1 ) < b r / >   if   α + 1 = - N ( n + α + 1 ) ⁢ ( n + α + β + 1 ) ⁢ ( n + γ + 1 ) ⁢ ( n - N ) ( 2 ⁢ n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β + 2 ) < b r / >   if   β + δ + 1 = - N ( n + α + 1 ) ⁢ ( n + α + β + 1 ) ⁢ ( n + β + δ + 1 ) ⁢ ( n - N ) ( 2 ⁢ n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β + 2 ) < b r / >   if   γ + 1 = - N subscript 𝐴 𝑛 𝑛 𝛼 1 𝑛 𝛼 𝛽 1 𝑛 𝛽 𝛿 1 𝑛 𝛾 1 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 2 cases 𝑛 𝛽 𝑁 𝑛 𝛽 𝛿 1 𝑛 𝛾 1 𝑛 𝑁 2 𝑛 𝛽 𝑁 2 𝑛 𝛽 𝑁 1 fragments b r italic-  if italic-  α 1 N 𝑛 𝛼 1 𝑛 𝛼 𝛽 1 𝑛 𝛾 1 𝑛 𝑁 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 2 fragments b r italic-  if italic-  β δ 1 N 𝑛 𝛼 1 𝑛 𝛼 𝛽 1 𝑛 𝛽 𝛿 1 𝑛 𝑁 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 2 fragments b r italic-  if italic-  γ 1 N {\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.}}} } &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Proof

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Symbols List

& : logical and
R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : Racah polynomial : http://dlmf.nist.gov/18.25#T1.t1.r4

Bibliography

Equation in Section 9.2 of KLS.

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