q-Racah

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q-Racah

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} {}=\qHyperrphis{4}{3}@@{q^{-n},\alpha\beta q^{n+1},q^{-x},\gamma\delta q^{x+1}}{\alpha q,\beta\delta q,\gamma q}{q}{q} }}

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 \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 q=q^{-N}\quad\textrm{or}\quad\beta\delta q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{q^{-x},\gamma\delta q^{x+1}}{q}{k}=\prod_{j=0}^{k-1}\left(1-\mu(x)q^j+\gamma\delta q^{2j+1}\right) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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


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{\qPochhammer{\alpha q,\beta\delta q,\gamma q,\gamma\delta q}{q}{x}} {\qPochhammer{q,\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q}{q}{x}} {} \frac{(1-\gamma\delta q^{2x+1})}{(\alpha\beta q)^x(1-\gamma\delta q)} \qRacah{m}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}\qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} =h_n\,\Kronecker{m}{n} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 h_n=\frac{\qPochhammer{\alpha^{-1}\beta^{-1}\gamma,\alpha^{-1}\delta,\beta^{-1},\gamma\delta q^2}{q}{\infty}} {\qPochhammer{\alpha^{-1}\beta^{-1}q^{-1},\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q}{q}{\infty}} {}\frac{(1-\alpha\beta q)(\gamma\delta q)^n}{(1-\alpha\beta q^{2n+1})} \frac{\qPochhammer{q,\alpha\beta\gamma^{-1}q,\alpha\delta^{-1}q,\beta q}{q}{n}}{\qPochhammer{\alpha q,\alpha\beta q,\beta\delta q,\gamma q}{q}{n}} =\left\{\begin{array}{ll} \displaystyle\frac{\qPochhammer{\beta^{-1},\gamma\delta q^2}{q}{N}}{\qPochhammer{\beta^{-1}\gamma q,\delta q}{q}{N}} \frac{(1-\beta q^{-N})(\gamma\delta q)^n}{(1-\beta q^{2n-N})} \frac{\qPochhammer{q,\beta q,\beta\gamma^{-1}q^{-N},\delta^{-1}q^{-N}}{q}{n}}{\qPochhammer{\beta q^{-N},\beta\delta q,\gamma q,q^{-N}}{q}{n}} &<br /> \quad\textrm{if}\quad\alpha q=q^{-N}\\ \\ \displaystyle\frac{\qPochhammer{\alpha\beta q^2,\beta\gamma^{-1}}{q}{N}}{\qPochhammer{\alpha\beta\gamma^{-1}q,\beta q}{q}{N}} \frac{(1-\alpha\beta q)(\beta^{-1}\gamma q^{-N})^n}{(1-\alpha\beta q^{2n+1})} \frac{\qPochhammer{q,\alpha\beta q^{N+2},\alpha\beta\gamma^{-1}q,\beta q}{q}{n}}{\qPochhammer{\alpha q,\alpha\beta q,\gamma q,q^{-N}}{q}{n}} &<br /> \quad\textrm{if}\quad\beta\delta q=q^{-N}\\ \\ \displaystyle\frac{\qPochhammer{\alpha\beta q^2,\delta^{-1}}{q}{N}}{\qPochhammer{\alpha\delta^{-1}q,\beta q}{q}{N}} \frac{(1-\alpha\beta q)(\delta q^{-N})^n}{(1-\alpha\beta q^{2n+1})} \frac{\qPochhammer{q,\alpha\beta q^{N+2},\alpha\delta^{-1}q,\beta q}{q}{n}} {\qPochhammer{\alpha q,\alpha\beta q,\beta\delta q,q^{-N}}{q}{n}} &<br /> \quad\textrm{if}\quad\gamma q=q^{-N} \end{array}\right.}} } &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}:=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 -\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)\qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} {}=A_n\qRacah{n+1}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}-\left(A_n+C_n\right)\qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}+C_n\qRacah{n-1}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 C_n=\frac{q(1-q^n)(1-\beta q^n)(\gamma-\alpha\beta q^n)(\delta-\alpha q^n)} {(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})(1-\beta\delta q^{n+1})(1-\gamma q^{n+1})} {(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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\monicqRacah{n}@@{x}{\alpha}{\beta}{\gamma}{\delta}{q}=\monicqRacah{n+1}@@{x}{\alpha}{\beta}{\gamma}{\delta}{q}+\left[1+\gamma\delta q-(A_n+C_n)\right]\monicqRacah{n}@@{x}{\alpha}{\beta}{\gamma}{\delta}{q}+A_{n-1}C_n\monicqRacah{n-1}@@{x}{\alpha}{\beta}{\gamma}{\delta}{q} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}= \frac{\qPochhammer{\alpha\beta q^{n+1}}{q}{n}}{\qPochhammer{\alpha q,\beta\delta q,\gamma q}{q}{n}}\monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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


q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Delta\left[w(x-1)B(x-1)\Delta y(x-1)\right] {}-q^{-n}(1-q^n)(1-\alpha\beta q^{n+1})w(x)y(x)=0 }}

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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;\alpha,\beta,\gamma,\delta|q)= \frac{\qPochhammer{\alpha q,\beta\delta q,\gamma q,\gamma\delta q}{q}{x}}{\qPochhammer{q,\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q}{q}{x}} \frac{(1-\gamma\delta q^{2x+1})}{(\alpha\beta q)^x(1-\gamma\delta q)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &
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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 q^{-n}(1-q^n)(1-\alpha\beta q^{n+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{q(1-q^x)(1-\delta q^x)(\beta-\gamma q^x)(\alpha-\gamma\delta q^x)} {(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=\frac{(1-\alpha q^{x+1})(1-\beta\delta q^{x+1})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})} {(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &
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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qRacah{n}@{\mu(x+1)}{\alpha}{\beta}{\gamma}{\delta}{q}-\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} {}=\frac{q^{-n-x}(1-q^n)(1-\alpha\beta q^{n+1})(1-\gamma\delta q^{2x+2})} {(1-\alpha q)(1-\beta\delta q)(1-\gamma q)} {} \qRacah{n-1}@{\mu(x)}{\alpha q}{\beta q}{\gamma q}{\delta}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}{\Delta\mu(x)} {}=\frac{q^{-n+1}(1-q^n)(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-\beta\delta q)(1-\gamma q)} {} \qRacah{n-1}@{\mu(x)}{\alpha q}{\beta q}{\gamma q}{\delta}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 (1-\alpha q^x)(1-\beta\delta q^x)(1-\gamma q^x)(1-\gamma\delta q^x)\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} {}-(1-q^x)(1-\delta q^x)(\alpha-\gamma\delta q^x)(\beta-\gamma q^x)\qRacah{n}@{\mu(x-1)}{\alpha}{\beta}{\gamma}{\delta}{q} {}=q^x(1-\alpha)(1-\beta\delta)(1-\gamma)(1-\gamma\delta q^{2x}) {} \qRacah{n+1}@{\mu(x)}{\alpha q^{-1}}{\beta q^{-1}}{\gamma q^{-1}}{\delta}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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[{\tilde w}(x;\alpha,\beta,\gamma,\delta|q)\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}\right]}{\nabla\mu(x)} {}=\frac{1}{(1-q)(1-\gamma\delta)}{\tilde w}(x;\alpha q^{-1},\beta q^{-1},\gamma q^{-1},\delta|q) {} \qRacah{n+1}@{\mu(x)}{\alpha q^{-1}}{\beta q^{-1}}{\gamma q^{-1}}{\delta}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 {\tilde w}(x;\alpha,\beta,\gamma,\delta|q)=\frac{\qPochhammer{\alpha q,\beta\delta q,\gamma q,\gamma\delta q}{q}{x}} {\qPochhammer{q,\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q}{q}{x}(\alpha\beta)^x} }}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;\alpha,\beta,\gamma,\delta|q)\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} {}=(1-q)^n\qPochhammer{\gamma\delta q}{q}{n}\left(\nabla_{\mu}\right)^n \left[{\tilde w}(x;\alpha q^n,\beta q^n,\gamma q^n,\delta|q)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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_{\mu}:=\frac{\nabla}{\nabla\mu(x)} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qHyperrphis{2}{1}@@{q^{-x},\alpha\gamma^{-1}\delta^{-1}q^{-x}}{\alpha q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\beta\delta q^{x+1},\gamma q^{x+1}}{\beta q}{q}{q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{\beta\delta q,\gamma q}{q}{n}}{\qPochhammer{\beta q,q}{q}{n}} \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}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 q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N}}}


Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qHyperrphis{2}{1}@@{q^{-x},\beta\gamma^{-1}q^{-x}}{\beta\delta q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\alpha q^{x+1},\gamma q^{x+1}}{\alpha\delta^{-1}q}{q}{q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{\alpha q,\gamma q}{q}{n}}{\qPochhammer{\alpha\delta^{-1}q,q}{q}{n}} \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}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 q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N}}}


Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qHyperrphis{2}{1}@@{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\alpha q^{x+1},\beta\delta q^{x+1}}{\alpha\beta\gamma^{-1}q}{q}{q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{\alpha q,\beta\delta q}{q}{n}}{\qPochhammer{\alpha\beta\gamma^{-1}q,q}{q}{n}} \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}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 q=q^{-N}\quad\textrm{or}\quad\beta\delta q=q^{-N}}}


Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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

q-Racah polynomial to Big q-Jacobi polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{a}{b}{c}{0}{q}=\bigqJacobi{n}@{q^{-x}}{a}{b}{c}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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


q-Racah polynomial to q-Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{q^{-N-1}}{0}{q}=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qRacah{n}@{\mu(x)}{\alpha}{\beta}{0}{\beta^{-1}q^{-N-1}}{q}=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qRacah{n}@{\mu(x)}{q^{-N-1}}{\beta\gamma q^{N+1}}{\gamma}{0}{q}=\qHahn{n}@{q^{-x}}{\gamma}{\beta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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


q-Racah polynomial to Dual q-Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{q^{-N-1}}{0}{\gamma}{\delta}{q}=\dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qRacah{n}@{\mu(x)}{0}{\delta^{-1}q^{-N-1}}{\gamma}{\delta}{q}=\dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \qRacah{n}@{\mu(x)}{\alpha}{0}{q^{-N-1}}{\alpha\delta q^{N+1}}{q}=\dualqHahn{n}@{{\tilde \mu}(x)}{\alpha}{\delta}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde \mu}(x)=q^{-x}+\alpha\delta q^{x+1}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &
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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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


q-Racah polynomial to q-Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{q^{-x}}{q^{-N-1}}{-pq^N}{0}{0}{q}=\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} }}

q-Racah polynomial to Dual q-Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{0}{0}{q^{-N-1}}{c}{q}=\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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


q-Racah polynomial to Racah polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\qRacah{n}@{\mu(x)}{q^{\alpha}}{q^{\beta}}{q^{\gamma}}{q^{\delta}}{q} =\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 \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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


Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{2a\cos@@{\theta}}{abq^{-1}}{cdq^{-1}}{adq^{-1}}{ad^{-1}}{q} =\frac{a^n\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab,ac,ad}{q}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q^{-1}}=\qRacah{n}@{{\tilde\mu}(x)}{\alpha^{-1}}{\beta^{-1}}{\gamma^{-1}}{\delta^{-1}}{q} }}

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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &
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 \mu(x):=q^{-x}+\gamma\delta q^{x+1}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}} &

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: q-Racah

Symmetry

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{x}{\alpha}{\beta}{q^{-N-1}}{\delta }{ q} =\frac{\qPochhammer{\beta q,\alpha\delta^{-1}q}{q}{n}}{\qPochhammer{\alpha q,\beta\delta q}{q}{n}} \delta^n \qRacah{n}@{\delta^{-1}x}{\beta}{\alpha}{q^{-N-1}}{\delta^{-1} }{ q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{x}{\alpha}{\beta}{q^{-N-1}}{-1 }{ q} =\frac{\qPochhammer{\beta q,-\alpha q}{q}{n}}{\qPochhammer{\alpha q,-\beta q}{q}{n}} (-1)^n \qRacah{n}@{-x}{\beta}{\alpha}{q^{-N-1}}{-1 }{ q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{x}{\alpha}{\alpha}{q^{-N-1}}{-1 }{ q} =(-1)^n \qRacah{n}@{-x}{\alpha}{\alpha}{q^{-N-1}}{-1 }{ q} }}

Trivial symmetry

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta }{ q}=\qRacah{n}@{x}{\beta\delta}{\alpha\delta^{-1}}{\gamma}{\delta }{ q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta }{ q} =\qRacah{n}@{x}{\gamma}{\alpha\beta\gamma^{-1}}{\alpha}{\gamma\delta\alpha^{-1} }{ q} }}

Duality

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{q^{-y}+\gamma\delta q^{y+1}}{q^{-N-1}}{\beta}{\gamma}{\delta }{ q} =\qRacah{y}@{q^{-n}+\beta q^{n-N}}{\gamma}{\delta}{q^{-N-1}}{\beta }{ q} (n,y }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{q^{-y}+\gamma\delta q^{y+1}}{q^{-N-1}}{\beta}{\gamma}{\delta }{ q} =0,1,\ldots,N) }}

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