Formula:KLS:09.02:38

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R n ⁡ ( x ⁢ ( x - N + δ ) ; α , β , - N - 1 , δ ) = W n ⁡ ( - ( x + 1 2 ⁢ ( δ - N ) ) 2 ; 1 2 ⁢ ( δ - N ) , α + 1 - 1 2 ⁢ ( δ - N ) , β + 1 2 ⁢ ( δ + N ) + 1 , - 1 2 ⁢ ( δ + N ) ) ( α + 1 ) n ⁢ ( β + δ + 1 ) n ⁢ ( - N ) n Racah-polynomial-R 𝑛 𝑥 𝑥 𝑁 𝛿 𝛼 𝛽 𝑁 1 𝛿 Wilson-polynomial-W 𝑛 superscript 𝑥 1 2 𝛿 𝑁 2 1 2 𝛿 𝑁 𝛼 1 1 2 𝛿 𝑁 𝛽 1 2 𝛿 𝑁 1 1 2 𝛿 𝑁 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 𝛿 1 𝑛 Pochhammer-symbol 𝑁 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(x(x-N+\delta);\alpha,% \beta,-N-1,\delta\right)=\frac{W_{n}\!\left(-(x+\frac{1}{2}(\delta-N))^{2};% \frac{1}{2}(\delta-N),\alpha+1-\frac{1}{2}(\delta-N),\beta+\frac{1}{2}(\delta+% N)+1,-\frac{1}{2}(\delta+N)\right)}{{\left(\alpha+1\right)_{n}}{\left(\beta+% \delta+1\right)_{n}}{\left(-N\right)_{n}}}}}}

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
W n subscript 𝑊 𝑛 {\displaystyle{\displaystyle{\displaystyle W_{n}}}}  : Wilson polynomial : http://dlmf.nist.gov/18.25#T1.t1.r2
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.2 of KLS.

URL links

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