Racah

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Racah

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n=0,1,2,\ldots,N}}


Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle M=\left\{\begin{array}{ll} \displaystyle\frac{\pochhammer{-\beta}{N}\pochhammer{\gamma+\delta+2}{N}}{\pochhammer{-\beta+\gamma+1}{N}\pochhammer{\delta+1}{N}} &<br /> \quad\textrm{if}\quad\alpha+1=-N\\[5mm] \displaystyle\frac{\pochhammer{-\alpha+\delta}{N}\pochhammer{\gamma+\delta+2}{N}}{\pochhammer{-\alpha+\gamma+\delta+1}{N}\pochhammer{\delta+1}{N}} &<br /> \quad\textrm{if}\quad\beta+\delta+1=-N\\[5mm] \displaystyle\frac{\pochhammer{\alpha+\beta+2}{N}\pochhammer{-\delta}{N}}{\pochhammer{\alpha-\delta+1}{N}\pochhammer{\beta+1}{N}} &<br /> \quad\textrm{if}\quad\gamma+1=-N \end{array}\right.}} } &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Racah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}:=\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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\\[5mm] \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\\[5mm] \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.}} } &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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\\[5mm] \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\\[5mm] \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.}} } &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Racah{n}@@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}:=\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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\\[5mm] \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\\[5mm] \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.}} } &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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\\[5mm] \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\\[5mm] \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.}} }


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D(x)=\frac{x(x-\alpha+\gamma+\delta)(x-\beta+\gamma)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \textrm{if}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N}}


Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \textrm{if}\quad\alpha+1=-N\quad\textrm{or}\quad\gamma+1=-N}}


Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \textrm{if}\quad\alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N}}


Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\pochhammer{\alpha+\beta+1}{n}}{n!}\Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{\delta}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Limit relations

Racah polynomial to Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\delta\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{-N-1}{\delta}=\Hahn{n}@{x}{\alpha}{\beta}{N} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\gamma\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{-\beta-N-1}=\Hahn{n}@{x}{\alpha}{\beta}{N} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Racah polynomial to Dual Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\beta\rightarrow\infty} \Racah{n}@{\lambda(x)}{-N-1}{\beta}{\gamma}{\delta}=\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{-\delta-N-1}{\gamma}{\delta}=\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Remark

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}


Koornwinder Addendum: Racah

Racah in terms of Wilson

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}

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