Racah

From DRMF
Jump to navigation Jump to search

Racah

Hypergeometric representation

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 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}}}} {\displaystyle \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} {}=\HyperpFq{4}{3}@@{-n,n+\alpha+\beta+1,-x,x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{1} }

Constraint(s): n = 0 , 1 , 2 , … , N 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}


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)}}}


α + 1 = - N   or   β + δ + 1 = - N   or   γ + 1 = - N formulae-sequence 𝛼 1 𝑁 or formulae-sequence 𝛽 𝛿 1 𝑁 or 𝛾 1 𝑁 {\displaystyle{\displaystyle{\displaystyle\alpha+1=-N\quad\textrm{or}\quad% \beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N}}} {\displaystyle \alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N }

Orthogonality relation(s)

∑ x = 0 N ( α + 1 ) x ⁢ ( β + δ + 1 ) x ⁢ ( γ + 1 ) x ⁢ ( γ + δ + 1 ) x ⁢ ( ( γ + δ + 3 ) / 2 ) x ( - α + γ + δ + 1 ) x ⁢ ( - β + γ + 1 ) x ⁢ ( ( γ + δ + 1 ) / 2 ) x ⁢ ( δ + 1 ) x ⁢ x ! ⁢ R m ⁡ ( λ ⁢ ( x ) ) ⁢ R n ⁡ ( λ ⁢ ( x ) ) = M ⁢ ( n + α + β + 1 ) n ⁢ ( α + β - γ + 1 ) n ⁢ ( α - δ + 1 ) n ⁢ ( β + 1 ) n ⁢ n ! ( α + β + 2 ) 2 ⁢ n ⁢ ( α + 1 ) n ⁢ ( β + δ + 1 ) n ⁢ ( γ + 1 ) n ⁢ δ m , n superscript subscript 𝑥 0 𝑁 Pochhammer-symbol 𝛼 1 𝑥 Pochhammer-symbol 𝛽 𝛿 1 𝑥 Pochhammer-symbol 𝛾 1 𝑥 Pochhammer-symbol 𝛾 𝛿 1 𝑥 Pochhammer-symbol 𝛾 𝛿 3 2 𝑥 Pochhammer-symbol 𝛼 𝛾 𝛿 1 𝑥 Pochhammer-symbol 𝛽 𝛾 1 𝑥 Pochhammer-symbol 𝛾 𝛿 1 2 𝑥 Pochhammer-symbol 𝛿 1 𝑥 𝑥 Racah-polynomial-R 𝑚 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 𝑀 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑛 Pochhammer-symbol 𝛼 𝛽 𝛾 1 𝑛 Pochhammer-symbol 𝛼 𝛿 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 𝑛 Pochhammer-symbol 𝛼 𝛽 2 2 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 𝛿 1 𝑛 Pochhammer-symbol 𝛾 1 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{{\left(\alpha+1% \right)_{x}}{\left(\beta+\delta+1\right)_{x}}{\left(\gamma+1\right)_{x}}{\left% (\gamma+\delta+1\right)_{x}}{\left((\gamma+\delta+3)/2\right)_{x}}}{{\left(-% \alpha+\gamma+\delta+1\right)_{x}}{\left(-\beta+\gamma+1\right)_{x}}{\left((% \gamma+\delta+1)/2\right)_{x}}{\left(\delta+1\right)_{x}}x!}{}R_{m}\!\left(% \lambda(x)\right)R_{n}\!\left(\lambda(x)\right){}=M\frac{{\left(n+\alpha+\beta% +1\right)_{n}}{\left(\alpha+\beta-\gamma+1\right)_{n}}{\left(\alpha-\delta+1% \right)_{n}}{\left(\beta+1\right)_{n}}n!}{{\left(\alpha+\beta+2\right)_{2n}}{% \left(\alpha+1\right)_{n}}{\left(\beta+\delta+1\right)_{n}}{\left(\gamma+1% \right)_{n}}}\,\delta_{m,n}}}} {\displaystyle \sum_{x=0}^N\frac{\pochhammer{\alpha+1}{x}\pochhammer{\beta+\delta+1}{x}\pochhammer{\gamma+1}{x}\pochhammer{\gamma+\delta+1}{x}\pochhammer{(\gamma+\delta+3)/2}{x}} {\pochhammer{-\alpha+\gamma+\delta+1}{x}\pochhammer{-\beta+\gamma+1}{x}\pochhammer{(\gamma+\delta+1)/2}{x}\pochhammer{\delta+1}{x}x!} {} \Racah{m}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}\Racah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} {}=M\frac{\pochhammer{n+\alpha+\beta+1}{n}\pochhammer{\alpha+\beta-\gamma+1}{n}\pochhammer{\alpha-\delta+1}{n}\pochhammer{\beta+1}{n}n!} {\pochhammer{\alpha+\beta+2}{2n}\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}\pochhammer{\gamma+1}{n}}\,\Kronecker{m}{n} }

Substitution(s): M = { ( - β ) N ⁢ ( γ + δ + 2 ) N ( - β + γ + 1 ) N ⁢ ( δ + 1 ) N < b r / >   if   α + 1 = - N ( - α + δ ) N ⁢ ( γ + δ + 2 ) N ( - α + γ + δ + 1 ) N ⁢ ( δ + 1 ) N < b r / >   if   β + δ + 1 = - N ( α + β + 2 ) N ⁢ ( - δ ) N ( α - δ + 1 ) N ⁢ ( β + 1 ) N < b r / >   if   γ + 1 = - N 𝑀 cases Pochhammer-symbol 𝛽 𝑁 Pochhammer-symbol 𝛾 𝛿 2 𝑁 Pochhammer-symbol 𝛽 𝛾 1 𝑁 Pochhammer-symbol 𝛿 1 𝑁 fragments b r italic-  if italic-  α 1 N Pochhammer-symbol 𝛼 𝛿 𝑁 Pochhammer-symbol 𝛾 𝛿 2 𝑁 Pochhammer-symbol 𝛼 𝛾 𝛿 1 𝑁 Pochhammer-symbol 𝛿 1 𝑁 fragments b r italic-  if italic-  β δ 1 N Pochhammer-symbol 𝛼 𝛽 2 𝑁 Pochhammer-symbol 𝛿 𝑁 Pochhammer-symbol 𝛼 𝛿 1 𝑁 Pochhammer-symbol 𝛽 1 𝑁 fragments b r italic-  if italic-  γ 1 N {\displaystyle{\displaystyle{\displaystyle M=\left\{\begin{array}[]{ll}% \displaystyle\frac{{\left(-\beta\right)_{N}}{\left(\gamma+\delta+2\right)_{N}}% }{{\left(-\beta+\gamma+1\right)_{N}}{\left(\delta+1\right)_{N}}}&<br/>\quad% \textrm{if}\quad\alpha+1=-N\\ \displaystyle\frac{{\left(-\alpha+\delta\right)_{N}}{\left(\gamma+\delta+2% \right)_{N}}}{{\left(-\alpha+\gamma+\delta+1\right)_{N}}{\left(\delta+1\right)% _{N}}}&<br/>\quad\textrm{if}\quad\beta+\delta+1=-N\\ \displaystyle\frac{{\left(\alpha+\beta+2\right)_{N}}{\left(-\delta\right)_{N}}% }{{\left(\alpha-\delta+1\right)_{N}}{\left(\beta+1\right)_{N}}}&<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)}}}


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

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)}}}


Recurrence relation

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)}}} {\displaystyle \lambda(x)\Racah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} =A_n\Racah{n+1}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}-\left(A_n+C_n\right)\Racah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}+C_n\Racah{n-1}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} }

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)}}}


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

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)}}}


Monic recurrence relation

x ⁢ R ^ n ⁡ ( x ) = R ^ n + 1 ⁡ ( x ) - ( A n + C n ) ⁢ R ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ R ^ n - 1 ⁡ ( x ) 𝑥 Racah-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 Racah-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝛾 𝛿 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Racah-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 Racah-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝛾 𝛿 {\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)}}} {\displaystyle x\monicRacah{n}@@{x}{\alpha}{\beta}{\gamma}{\delta}=\monicRacah{n+1}@@{x}{\alpha}{\beta}{\gamma}{\delta}-(A_n+C_n)\monicRacah{n}@@{x}{\alpha}{\beta}{\gamma}{\delta}+A_{n-1}C_n\monicRacah{n-1}@@{x}{\alpha}{\beta}{\gamma}{\delta} }

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.}}} }


R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) = ( n + α + β + 1 ) n ( α + 1 ) n ⁢ ( β + δ + 1 ) n ⁢ ( γ + 1 ) n ⁢ R ^ n ⁡ ( λ ⁢ ( x ) ) Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 𝛿 1 𝑛 Pochhammer-symbol 𝛾 1 𝑛 Racah-polynomial-monic-p 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\lambda(x);\alpha,% \beta,\gamma,\delta\right)=\frac{{\left(n+\alpha+\beta+1\right)_{n}}}{{\left(% \alpha+1\right)_{n}}{\left(\beta+\delta+1\right)_{n}}{\left(\gamma+1\right)_{n% }}}{\widehat{R}}_{n}\!\left(\lambda(x)\right)}}} {\displaystyle \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}= \frac{\pochhammer{n+\alpha+\beta+1}{n}}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}\pochhammer{\gamma+1}{n}}\monicRacah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} }

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)}}}


Difference equation

n ⁢ ( n + α + β + 1 ) ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( x + 1 ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( x - 1 ) 𝑛 𝑛 𝛼 𝛽 1 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 1 delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 1 {\displaystyle{\displaystyle{\displaystyle n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-% \left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}} {\displaystyle n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1) }

Substitution(s): D ⁢ ( x ) = x ⁢ ( x - α + γ + δ ) ⁢ ( x - β + γ ) ⁢ ( x + δ ) ( 2 ⁢ x + γ + δ ) ⁢ ( 2 ⁢ x + γ + δ + 1 ) 𝐷 𝑥 𝑥 𝑥 𝛼 𝛾 𝛿 𝑥 𝛽 𝛾 𝑥 𝛿 2 𝑥 𝛾 𝛿 2 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle D(x)=\frac{x(x-\alpha+\gamma+\delta% )(x-\beta+\gamma)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}}}} &

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


Forward shift operator

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

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)}}}


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

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)}}}


Backward shift operator

( 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)}}} {\displaystyle (x+\alpha)(x+\beta+\delta)(x+\gamma)(x+\gamma+\delta)\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} {}-x(x-\beta+\gamma)(x-\alpha+\gamma+\delta)(x+\delta)\Racah{n}@{\lambda(x-1)}{\alpha}{\beta}{\gamma}{\delta} {}=\alpha\gamma(\beta+\delta)(2x+\gamma+\delta)\Racah{n+1}@{\lambda(x)}{\alpha-1}{\beta-1}{\gamma-1}{\delta} }

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)}}}


∇ ⁡ [ ω ⁢ ( x ; α , β , γ , δ ) ⁢ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) ] ∇ ⁡ λ ⁢ ( x ) = 1 γ + δ ⁢ ω ⁢ ( x ; α - 1 , β - 1 , γ - 1 , δ ) ⁢ R n + 1 ⁡ ( λ ⁢ ( x ) ; α - 1 , β - 1 , γ - 1 , δ ) ∇ 𝜔 𝑥 𝛼 𝛽 𝛾 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 ∇ 𝜆 𝑥 1 𝛾 𝛿 𝜔 𝑥 𝛼 1 𝛽 1 𝛾 1 𝛿 Racah-polynomial-R 𝑛 1 𝜆 𝑥 𝛼 1 𝛽 1 𝛾 1 𝛿 {\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[\omega(x;\alpha,% \beta,\gamma,\delta)R_{n}\!\left(\lambda(x);\alpha,\beta,\gamma,\delta\right)% \right]}{\nabla\lambda(x)}{}=\frac{1}{\gamma+\delta}\omega(x;\alpha-1,\beta-1,% \gamma-1,\delta)R_{n+1}\!\left(\lambda(x);\alpha-1,\beta-1,\gamma-1,\delta% \right)}}} {\displaystyle \frac{\nabla\left[\omega(x;\alpha,\beta,\gamma,\delta)\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}\right]}{\nabla\lambda(x)} {}=\frac{1}{\gamma+\delta}\omega(x;\alpha-1,\beta-1,\gamma-1,\delta) \Racah{n+1}@{\lambda(x)}{\alpha-1}{\beta-1}{\gamma-1}{\delta} }

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)}}}


ω ⁢ ( x ; α , β , γ , δ ) = ( α + 1 ) x ⁢ ( β + δ + 1 ) x ⁢ ( γ + 1 ) x ⁢ ( γ + δ + 1 ) x ( - α + γ + δ + 1 ) x ⁢ ( - β + γ + 1 ) x ⁢ ( δ + 1 ) x ⁢ x ! 𝜔 𝑥 𝛼 𝛽 𝛾 𝛿 Pochhammer-symbol 𝛼 1 𝑥 Pochhammer-symbol 𝛽 𝛿 1 𝑥 Pochhammer-symbol 𝛾 1 𝑥 Pochhammer-symbol 𝛾 𝛿 1 𝑥 Pochhammer-symbol 𝛼 𝛾 𝛿 1 𝑥 Pochhammer-symbol 𝛽 𝛾 1 𝑥 Pochhammer-symbol 𝛿 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,\gamma,\delta)% =\frac{{\left(\alpha+1\right)_{x}}{\left(\beta+\delta+1\right)_{x}}{\left(% \gamma+1\right)_{x}}{\left(\gamma+\delta+1\right)_{x}}}{{\left(-\alpha+\gamma+% \delta+1\right)_{x}}{\left(-\beta+\gamma+1\right)_{x}}{\left(\delta+1\right)_{% x}}x!}}}} {\displaystyle \omega(x;\alpha,\beta,\gamma,\delta)=\frac{\pochhammer{\alpha+1}{x}\pochhammer{\beta+\delta+1}{x}\pochhammer{\gamma+1}{x}\pochhammer{\gamma+\delta+1}{x}} {\pochhammer{-\alpha+\gamma+\delta+1}{x}\pochhammer{-\beta+\gamma+1}{x}\pochhammer{\delta+1}{x}x!} }

Rodrigues-type formula

ω ⁢ ( x ; α , β , γ , δ ) ⁢ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) = ( γ + δ + 1 ) n ⁢ ( ∇ λ ) n ⁢ [ ω ⁢ ( x ; α + n , β + n , γ + n , δ ) ] 𝜔 𝑥 𝛼 𝛽 𝛾 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 Pochhammer-symbol 𝛾 𝛿 1 𝑛 superscript subscript ∇ 𝜆 𝑛 delimited-[] 𝜔 𝑥 𝛼 𝑛 𝛽 𝑛 𝛾 𝑛 𝛿 {\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,\gamma,\delta)% R_{n}\!\left(\lambda(x);\alpha,\beta,\gamma,\delta\right){}={\left(\gamma+% \delta+1\right)_{n}}\left(\nabla_{\lambda}\right)^{n}\left[\omega(x;\alpha+n,% \beta+n,\gamma+n,\delta)\right]}}} {\displaystyle \omega(x;\alpha,\beta,\gamma,\delta)\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} {}=\pochhammer{\gamma+\delta+1}{n}\left(\nabla_{\lambda}\right)^n\left[\omega(x;\alpha+n,\beta+n,\gamma+n,\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)}}}


∇ λ := ∇ ∇ ⁡ λ ⁢ ( x ) assign subscript ∇ 𝜆 ∇ ∇ 𝜆 𝑥 {\displaystyle{\displaystyle{\displaystyle\nabla_{\lambda}:=\frac{\nabla}{% \nabla\lambda(x)}}}} {\displaystyle \nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)} }

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)}}}


Generating functions

\HyperpFq ⁢ 21 ⁢ @ ⁢ @ - x , - x + α - γ - δ ⁢ α + 1 ⁢ t ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ x + β + δ + 1 , x + γ + 1 ⁢ β + 1 ⁢ t = ∑ n = 0 N ( β + δ + 1 ) n ⁢ ( γ + 1 ) n ( β + 1 ) n ⁢ n ! ⁢ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) ⁢ t n \HyperpFq 21 @ @ 𝑥 𝑥 𝛼 𝛾 𝛿 𝛼 1 𝑡 \HyperpFq 21 @ @ 𝑥 𝛽 𝛿 1 𝑥 𝛾 1 𝛽 1 𝑡 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛽 𝛿 1 𝑛 Pochhammer-symbol 𝛾 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 𝑛 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{-x,-x+\alpha-% \gamma-\delta}{\alpha+1}{t}\,\HyperpFq{2}{1}@@{x+\beta+\delta+1,x+\gamma+1}{% \beta+1}{t}{}=\sum_{n=0}^{N}\frac{{\left(\beta+\delta+1\right)_{n}}{\left(% \gamma+1\right)_{n}}}{{\left(\beta+1\right)_{n}}n!}R_{n}\!\left(\lambda(x);% \alpha,\beta,\gamma,\delta\right)t^{n}{}}}} {\displaystyle \HyperpFq{2}{1}@@{-x,-x+\alpha-\gamma-\delta}{\alpha+1}{t}\,\HyperpFq{2}{1}@@{x+\beta+\delta+1,x+\gamma+1}{\beta+1}{t} {}=\sum_{n=0}^N\frac{\pochhammer{\beta+\delta+1}{n}\pochhammer{\gamma+1}{n}}{\pochhammer{\beta+1}{n}n!} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}t^n {} }

Constraint(s): if   β + δ + 1 = - N   or   γ + 1 = - N formulae-sequence if 𝛽 𝛿 1 𝑁 or 𝛾 1 𝑁 {\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\beta+\delta+1=-N% \quad\textrm{or}\quad\gamma+1=-N}}}


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)}}}


\HyperpFq ⁢ 21 ⁢ @ ⁢ @ - x , - x + β - γ ⁢ β + δ + 1 ⁢ t ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ x + α + 1 , x + γ + 1 ⁢ α - δ + 1 ⁢ t = ∑ n = 0 N ( α + 1 ) n ⁢ ( γ + 1 ) n ( α - δ + 1 ) n ⁢ n ! ⁢ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) ⁢ t n \HyperpFq 21 @ @ 𝑥 𝑥 𝛽 𝛾 𝛽 𝛿 1 𝑡 \HyperpFq 21 @ @ 𝑥 𝛼 1 𝑥 𝛾 1 𝛼 𝛿 1 𝑡 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛾 1 𝑛 Pochhammer-symbol 𝛼 𝛿 1 𝑛 𝑛 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{-x,-x+\beta-\gamma% }{\beta+\delta+1}{t}\,\HyperpFq{2}{1}@@{x+\alpha+1,x+\gamma+1}{\alpha-\delta+1% }{t}{}=\sum_{n=0}^{N}\frac{{\left(\alpha+1\right)_{n}}{\left(\gamma+1\right)_{% n}}}{{\left(\alpha-\delta+1\right)_{n}}n!}R_{n}\!\left(\lambda(x);\alpha,\beta% ,\gamma,\delta\right)t^{n}{}}}} {\displaystyle \HyperpFq{2}{1}@@{-x,-x+\beta-\gamma}{\beta+\delta+1}{t}\,\HyperpFq{2}{1}@@{x+\alpha+1,x+\gamma+1}{\alpha-\delta+1}{t} {}=\sum_{n=0}^N\frac{\pochhammer{\alpha+1}{n}\pochhammer{\gamma+1}{n}}{\pochhammer{\alpha-\delta+1}{n}n!} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}t^n {} }

Constraint(s): if   α + 1 = - N   or   γ + 1 = - N formulae-sequence if 𝛼 1 𝑁 or 𝛾 1 𝑁 {\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\alpha+1=-N\quad% \textrm{or}\quad\gamma+1=-N}}}


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)}}}


\HyperpFq ⁢ 21 ⁢ @ ⁢ @ - x , - x - δ ⁢ γ + 1 ⁢ t ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ x + α + 1 , x + β + δ + 1 ⁢ α + β - γ + 1 ⁢ t = ∑ n = 0 N ( α + 1 ) n ⁢ ( β + δ + 1 ) n ( α + β - γ + 1 ) n ⁢ n ! ⁢ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) ⁢ t n \HyperpFq 21 @ @ 𝑥 𝑥 𝛿 𝛾 1 𝑡 \HyperpFq 21 @ @ 𝑥 𝛼 1 𝑥 𝛽 𝛿 1 𝛼 𝛽 𝛾 1 𝑡 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 𝛿 1 𝑛 Pochhammer-symbol 𝛼 𝛽 𝛾 1 𝑛 𝑛 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{-x,-x-\delta}{% \gamma+1}{t}\,\HyperpFq{2}{1}@@{x+\alpha+1,x+\beta+\delta+1}{\alpha+\beta-% \gamma+1}{t}{}=\sum_{n=0}^{N}\frac{{\left(\alpha+1\right)_{n}}{\left(\beta+% \delta+1\right)_{n}}}{{\left(\alpha+\beta-\gamma+1\right)_{n}}n!}R_{n}\!\left(% \lambda(x);\alpha,\beta,\gamma,\delta\right)t^{n}{}}}} {\displaystyle \HyperpFq{2}{1}@@{-x,-x-\delta}{\gamma+1}{t}\,\HyperpFq{2}{1}@@{x+\alpha+1,x+\beta+\delta+1}{\alpha+\beta-\gamma+1}{t} {}=\sum_{n=0}^N\frac{\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}}{\pochhammer{\alpha+\beta-\gamma+1}{n}n!} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}t^n {} }

Constraint(s): if   α + 1 = - N   or   β + δ + 1 = - N formulae-sequence if 𝛼 1 𝑁 or 𝛽 𝛿 1 𝑁 {\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\alpha+1=-N\quad% \textrm{or}\quad\beta+\delta+1=-N}}}


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)}}}


[ ( 1 - t ) - α - β - 1 ⁢ \HyperpFq ⁢ 43 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ ( α + β + 1 ) , 1 2 ⁢ ( α + β + 2 ) , - x , x + γ + δ + 1 ⁢ α + 1 , β + δ + 1 , γ + 1 - 4 ⁢ t ( 1 - t ) 2 ] N = ∑ n = 0 N ( α + β + 1 ) n n ! ⁢ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) ⁢ t n subscript superscript 1 𝑡 𝛼 𝛽 1 \HyperpFq 43 @ @ 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝑥 𝑥 𝛾 𝛿 1 𝛼 1 𝛽 𝛿 1 𝛾 1 4 𝑡 superscript 1 𝑡 2 𝑁 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛼 𝛽 1 𝑛 𝑛 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\Bigg{[}(1-t)^{-\alpha-\beta-1}{}% \HyperpFq{4}{3}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x,x% +\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{-\frac{4t}{(1-t)^{2}}}% \Bigg{]}_{N}{}=\sum_{n=0}^{N}\frac{{\left(\alpha+\beta+1\right)_{n}}}{n!}R_{n}% \!\left(\lambda(x);\alpha,\beta,\gamma,\delta\right)t^{n}}}} {\displaystyle \Bigg[(1-t)^{-\alpha-\beta-1} {}\HyperpFq{4}{3}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x,x+\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{-\frac{4t}{(1-t)^2}}\Bigg]_N {}=\sum_{n=0}^N\frac{\pochhammer{\alpha+\beta+1}{n}}{n!}\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}t^n }

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)}}}


Limit relations

Racah polynomial to Hahn polynomial

lim δ → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; α , β , - N - 1 , δ ) = Q n ⁡ ( x ; α , β , N ) subscript → 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝑁 1 𝛿 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\delta\rightarrow\infty}R_{n}% \!\left(\lambda(x);\alpha,\beta,-N-1,\delta\right)=Q_{n}\!\left(x;\alpha,\beta% ,N\right)}}} {\displaystyle \lim_{\delta\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{-N-1}{\delta}=\Hahn{n}@{x}{\alpha}{\beta}{N} }

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)}}}


lim γ → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , - β - N - 1 ) = Q n ⁡ ( x ; α , β , N ) subscript → 𝛾 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛽 𝑁 1 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\gamma\rightarrow\infty}R_{n}% \!\left(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1\right)=Q_{n}\!\left(x;\alpha% ,\beta,N\right)}}} {\displaystyle \lim_{\gamma\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{-\beta-N-1}=\Hahn{n}@{x}{\alpha}{\beta}{N} }

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)}}}


lim δ → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; - N - 1 , β + γ + N + 1 , γ , δ ) = Q n ⁡ ( x ; γ , β , N ) subscript → 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝑁 1 𝛽 𝛾 𝑁 1 𝛾 𝛿 Hahn-polynomial-Q 𝑛 𝑥 𝛾 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\delta\rightarrow\infty}R_{n}% \!\left(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta\right)=Q_{n}\!\left(x;% \gamma,\beta,N\right)}}} {\displaystyle \lim_{\delta\rightarrow\infty} \Racah{n}@{\lambda(x)}{-N-1}{\beta+\gamma+N+1}{\gamma}{\delta}=\Hahn{n}@{x}{\gamma}{\beta}{N} }

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)}}}


Racah polynomial to Dual Hahn polynomial

lim β → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; - N - 1 , β , γ , δ ) = R n ⁡ ( λ ⁢ ( x ) ; γ , δ , N ) subscript → 𝛽 Racah-polynomial-R 𝑛 𝜆 𝑥 𝑁 1 𝛽 𝛾 𝛿 dual-Hahn-R 𝑛 𝜆 𝑥 𝛾 𝛿 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}R_{n}\!% \left(\lambda(x);-N-1,\beta,\gamma,\delta\right)=R_{n}\!\left(\lambda(x);% \gamma,\delta,N\right)}}} {\displaystyle \lim_{\beta\rightarrow\infty} \Racah{n}@{\lambda(x)}{-N-1}{\beta}{\gamma}{\delta}=\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N} }

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)}}}


lim α → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; α , - δ - N - 1 , γ , δ ) = R n ⁡ ( λ ⁢ ( x ) ; γ , δ , N ) subscript → 𝛼 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛿 𝑁 1 𝛾 𝛿 dual-Hahn-R 𝑛 𝜆 𝑥 𝛾 𝛿 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}R_{n}% \!\left(\lambda(x);\alpha,-\delta-N-1,\gamma,\delta\right)=R_{n}\!\left(% \lambda(x);\gamma,\delta,N\right)}}} {\displaystyle \lim_{\alpha\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{-\delta-N-1}{\gamma}{\delta}=\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N} }

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)}}}


lim β → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; α , β , - N - 1 , α + δ + N + 1 ) = R n ⁡ ( λ ⁢ ( x ) ; α , δ , N ) subscript → 𝛽 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝑁 1 𝛼 𝛿 𝑁 1 dual-Hahn-R 𝑛 𝜆 𝑥 𝛼 𝛿 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}R_{n}\!% \left(\lambda(x);\alpha,\beta,-N-1,\alpha+\delta+N+1\right)=R_{n}\!\left(% \lambda(x);\alpha,\delta,N\right)}}} {\displaystyle \lim_{\beta\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{-N-1}{\alpha+\delta+N+1}=\dualHahn{n}@{\lambda(x)}{\alpha}{\delta}{N} }

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)}}}


Remark

R n ⁡ ( λ ⁢ ( - a + i ⁢ x ) ; a + b - 1 , c + d - 1 , a + d - 1 , a - d ) = W ~ n ⁡ ( x 2 ; a , b , c , d ) Racah-polynomial-R 𝑛 𝜆 𝑎 imaginary-unit 𝑥 𝑎 𝑏 1 𝑐 𝑑 1 𝑎 𝑑 1 𝑎 𝑑 Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\lambda(-a+\mathrm{i}x% );a+b-1,c+d-1,a+d-1,a-d\right){}={\tilde{W}}_{n}\!\left(x^{2};a,b,c,d\right)}}} {\displaystyle \Racah{n}@{\lambda(-a+\iunit x)}{a+b-1}{c+d-1}{a+d-1}{a-d} {}=\normWilsonWtilde{n}@{x^2}{a}{b}{c}{d} }

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)}}}


R n ⁡ ( λ ⁢ ( - a + i ⁢ x ) ; a + b - 1 , c + d - 1 , a + d - 1 , a - d ) = W n ⁡ ( x 2 ; a , b , c , d ) ( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n Racah-polynomial-R 𝑛 𝜆 𝑎 imaginary-unit 𝑥 𝑎 𝑏 1 𝑐 𝑑 1 𝑎 𝑑 1 𝑎 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\lambda(-a+\mathrm{i}x% );a+b-1,c+d-1,a+d-1,a-d\right){}=\frac{W_{n}\!\left(x^{2};a,b,c,d\right)}{{% \left(a+b\right)_{n}}{\left(a+c\right)_{n}}{\left(a+d\right)_{n}}}}}} {\displaystyle \Racah{n}@{\lambda(-a+\iunit x)}{a+b-1}{c+d-1}{a+d-1}{a-d} {}=\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}} }

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)}}}


Koornwinder Addendum: Racah

Racah in terms of Wilson

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}}}}}} {\displaystyle \Racah{n}@{x(x-N+\delta)}{\alpha}{\beta}{-N-1}{\delta} =\frac{\Wilson{n}@{-(x+\frac12(\delta-N))^2}{\frac12(\delta-N)}{\alpha+1-\frac12(\delta-N)}{\beta+\frac12(\delta+N)+1}{-\frac12(\delta+N)}} {\pochhammer{\alpha+1}{n} \pochhammer{\beta+\delta+1}{n} \pochhammer{-N}{n}} }

Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Racah&oldid=1466"