q-Racah

<< Askey-Wilson

q-Racah

Continuous dual q-Hahn >>

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}{}=\qHyperrphis{4}{3}@@{q^{-n},\alpha\beta q^{n+1},q^{-x},\gamma\delta q^{x+1}}{\alpha q,\beta\delta q,\gamma q}{q}{q}}}}$

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

${\displaystyle{\displaystyle{\displaystyle\alpha q=q^{-N}\quad\textrm{or}\quad\beta\delta q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(q^{-x},\gamma\delta q^{x+1};q\right)_{k}=\prod_{j=0}^{k-1}\left(1-\mu(x)q^{j}+\gamma\delta q^{2j+1}\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\qPochhammer{\alpha q,\beta\delta q,\gamma q,\gamma\delta q}{q}{x}}{\qPochhammer{q,\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q}{q}{x}}{}\frac{(1-\gamma\delta q^{2x+1})}{(\alpha\beta q)^{x}(1-\gamma\delta q)}\qRacah{m}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}\qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}=h_{n}\,\Kronecker{m}{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)\qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}{}=A_{n}\qRacah{n+1}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}-\left(A_{n}+C_{n}\right)\qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}+C_{n}\qRacah{n-1}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{q(1-q^{n})(1-\beta q^{n})(\gamma-\alpha\beta q^{n})(\delta-\alpha q^{n})}{(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})(1-\beta\delta q^{n+1})(1-\gamma q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

Monic recurrence relation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{q(1-q^{n})(1-\beta q^{n})(\gamma-\alpha\beta q^{n})(\delta-\alpha q^{n})}{(1-\alpha\beta q^{2n})(1-\alpha\beta q^{2n+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-\alpha q^{n+1})(1-\alpha\beta q^{n+1})(1-\beta\delta q^{n+1})(1-\gamma q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2})}}}}$

${\displaystyle{\displaystyle{\displaystyle\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}=\frac{\qPochhammer{\alpha\beta q^{n+1}}{q}{n}}{\qPochhammer{\alpha q,\beta\delta q,\gamma q}{q}{n}}\monicqRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle\Delta\left[w(x-1)B(x-1)\Delta y(x-1)\right]{}-q^{-n}(1-q^{n})(1-\alpha\beta q^{n+1})w(x)y(x)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(1-\alpha q^{x+1})(1-\beta\delta q^{x+1})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})}{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}}}}$ &

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;\alpha,\beta,\gamma,\delta|q)=\frac{\qPochhammer{\alpha q,\beta\delta q,\gamma q,\gamma\delta q}{q}{x}}{\qPochhammer{q,\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q}{q}{x}}\frac{(1-\gamma\delta q^{2x+1})}{(\alpha\beta q)^{x}(1-\gamma\delta q)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle y(x)=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

${\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})(1-\alpha\beta q^{n+1})y(x){}=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle D(x)=\frac{q(1-q^{x})(1-\delta q^{x})(\beta-\gamma q^{x})(\alpha-\gamma\delta q^{x})}{(1-\gamma\delta q^{2x})(1-\gamma\delta q^{2x+1})}}}}$ &

${\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(1-\alpha q^{x+1})(1-\beta\delta q^{x+1})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})}{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle y(x)=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

Forward shift operator

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

Backward shift operator

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;\alpha,\beta,\gamma,\delta|q)=\frac{\left(\alpha q,\beta\delta q,\gamma q,\gamma\delta q;q\right)_{x}}{\left(q,\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q;q\right)_{x}(\alpha\beta)^{x}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;\alpha,\beta,\gamma,\delta|q)\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}{}=(1-q)^{n}\qPochhammer{\gamma\delta q}{q}{n}\left(\nabla_{\mu}\right)^{n}\left[{\tilde{w}}(x;\alpha q^{n},\beta q^{n},\gamma q^{n},\delta|q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &