Racah

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Racah

Hypergeometric representation

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Orthogonality relation(s)

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Recurrence relation

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.}} } &


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Monic recurrence relation

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.}} }


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Difference equation

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Forward shift operator

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Backward shift operator

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Rodrigues-type formula

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Generating functions

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Limit relations

Racah polynomial to Hahn polynomial

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Racah polynomial to Dual Hahn polynomial

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Remark

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Koornwinder Addendum: Racah

Racah in terms of Wilson