Definition:Racah

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The LaTeX DLMF and DRMF macro \Racah represents the Racah polynomial.

This macro is in the category of polynomials.

In math mode, this macro can be called in the following ways:

\Racah{n} produces R n Racah-polynomial-R 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}
\Racah{n}@{x}{\alpha}{\beta}{\gamma}{\delta} produces R n ⁡ ( x ; α , β , γ , δ ) Racah-polynomial-R 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(x;\alpha,\beta,\gamma,% \delta\right)}}}

These are defined by R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) := \HyperpFq ⁢ 43 ⁢ @ ⁢ @ - n , n + α + β + 1 , - x , x + γ + δ + 1 ⁢ α + 1 , β + δ + 1 , γ + 11 , assign Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 \HyperpFq 43 @ @ 𝑛 𝑛 𝛼 𝛽 1 𝑥 𝑥 𝛾 𝛿 1 𝛼 1 𝛽 𝛿 1 𝛾 11 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\lambda(x);\alpha,% \beta,\gamma,\delta\right){}:=\HyperpFq{4}{3}@@{-n,n+\alpha+\beta+1,-x,x+% \gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1},}}}

where λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) , 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1),}}} and α + 1 = - N 𝛼 1 𝑁 {\displaystyle{\displaystyle\alpha+1=-N}} or β + δ + 1 = - N 𝛽 𝛿 1 𝑁 {\displaystyle{\displaystyle\beta+\delta+1=-N}} or γ + 1 = - N 𝛾 1 𝑁 {\displaystyle{\displaystyle\gamma+1=-N}} with N 𝑁 {\displaystyle{\displaystyle N}} a nonnegative integer.

Symbols List

R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : Racah polynomial : http://dlmf.nist.gov/18.25#T1.t1.r4
F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1


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