Formula:KLS:09.02:21

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( x + α ) ⁢ ( x + β + δ ) ⁢ ( x + γ ) ⁢ ( x + γ + δ ) ⁢ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) - x ⁢ ( x - β + γ ) ⁢ ( x - α + γ + δ ) ⁢ ( x + δ ) ⁢ R n ⁡ ( λ ⁢ ( x - 1 ) ; α , β , γ , δ ) = α ⁢ γ ⁢ ( β + δ ) ⁢ ( 2 ⁢ x + γ + δ ) ⁢ R n + 1 ⁡ ( λ ⁢ ( x ) ; α - 1 , β - 1 , γ - 1 , δ ) 𝑥 𝛼 𝑥 𝛽 𝛿 𝑥 𝛾 𝑥 𝛾 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 𝑥 𝑥 𝛽 𝛾 𝑥 𝛼 𝛾 𝛿 𝑥 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 1 𝛼 𝛽 𝛾 𝛿 𝛼 𝛾 𝛽 𝛿 2 𝑥 𝛾 𝛿 Racah-polynomial-R 𝑛 1 𝜆 𝑥 𝛼 1 𝛽 1 𝛾 1 𝛿 {\displaystyle{\displaystyle{\displaystyle(x+\alpha)(x+\beta+\delta)(x+\gamma)% (x+\gamma+\delta)R_{n}\!\left(\lambda(x);\alpha,\beta,\gamma,\delta\right){}-x% (x-\beta+\gamma)(x-\alpha+\gamma+\delta)(x+\delta)R_{n}\!\left(\lambda(x-1);% \alpha,\beta,\gamma,\delta\right){}=\alpha\gamma(\beta+\delta)(2x+\gamma+% \delta)R_{n+1}\!\left(\lambda(x);\alpha-1,\beta-1,\gamma-1,\delta\right)}}}

Substitution(s)

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

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.

URL links

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