q-Racah

<< Askey-Wilson

q-Racah

Continuous dual q-Hahn >>

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right){}=\qHyperrphis{4}{3}@@{q^{-n},\alpha\beta q^{n+1},q^{-x},\gamma\delta q^{x+1}}{\alpha q,\beta\delta q,\gamma q}{q}{q}}}}$

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)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(\alpha q,\beta\delta q,\gamma q,\gamma\delta q;q\right)_{x}}{\left(q,\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q;q\right)_{x}}{}\frac{(1-\gamma\delta q^{2x+1})}{(\alpha\beta q)^{x}(1-\gamma\delta q)}R_{m}\!\left(\mu(x)\right)R_{n}\!\left(\mu(x)\right)=h_{n}\,\delta_{m,n}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{\left(\alpha^{-1}\beta^{-1}\gamma,\alpha^{-1}\delta,\beta^{-1},\gamma\delta q^{2};q\right)_{\infty}}{\left(\alpha^{-1}\beta^{-1}q^{-1},\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q;q\right)_{\infty}}{}\frac{(1-\alpha\beta q)(\gamma\delta q)^{n}}{(1-\alpha\beta q^{2n+1})}\frac{\left(q,\alpha\beta\gamma^{-1}q,\alpha\delta^{-1}q,\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,\beta\delta q,\gamma q;q\right)_{n}}=\left\{\begin{array}[]{ll}\displaystyle\frac{\left(\beta^{-1},\gamma\delta q^{2};q\right)_{N}}{\left(\beta^{-1}\gamma q,\delta q;q\right)_{N}}\frac{(1-\beta q^{-N})(\gamma\delta q)^{n}}{(1-\beta q^{2n-N})}\frac{\left(q,\beta q,\beta\gamma^{-1}q^{-N},\delta^{-1}q^{-N};q\right)_{n}}{\left(\beta q^{-N},\beta\delta q,\gamma q,q^{-N};q\right)_{n}}&<br/>\quad\textrm{if}\quad\alpha q=q^{-N}\\ \\ \displaystyle\frac{\left(\alpha\beta q^{2},\beta\gamma^{-1};q\right)_{N}}{\left(\alpha\beta\gamma^{-1}q,\beta q;q\right)_{N}}\frac{(1-\alpha\beta q)(\beta^{-1}\gamma q^{-N})^{n}}{(1-\alpha\beta q^{2n+1})}\frac{\left(q,\alpha\beta q^{N+2},\alpha\beta\gamma^{-1}q,\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,\gamma q,q^{-N};q\right)_{n}}&<br/>\quad\textrm{if}\quad\beta\delta q=q^{-N}\\ \\ \displaystyle\frac{\left(\alpha\beta q^{2},\delta^{-1};q\right)_{N}}{\left(\alpha\delta^{-1}q,\beta q;q\right)_{N}}\frac{(1-\alpha\beta q)(\delta q^{-N})^{n}}{(1-\alpha\beta q^{2n+1})}\frac{\left(q,\alpha\beta q^{N+2},\alpha\delta^{-1}q,\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,\beta\delta q,q^{-N};q\right)_{n}}&<br/>\quad\textrm{if}\quad\gamma q=q^{-N}\end{array}\right.}}}$ } &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x)\right):=R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)}}}$

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

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

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

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-\left(1-q^{-x}\right)\left(1-\gamma\delta q^{x+1}\right)R_{n}\!\left(\mu(x)\right){}=A_{n}R_{n+1}\!\left(\mu(x)\right)-\left(A_{n}+C_{n}\right)R_{n}\!\left(\mu(x)\right)+C_{n}R_{n-1}\!\left(\mu(x)\right)}}}$

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

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

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

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{R}}_{n}\!\left(x\right)={\widehat{R}}_{n+1}\!\left(x\right)+\left[1+\gamma\delta q-(A_{n}+C_{n})\right]{\widehat{R}}_{n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{R}}_{n-1}\!\left(x\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)=\frac{\left(\alpha\beta q^{n+1};q\right)_{n}}{\left(\alpha q,\beta\delta q,\gamma q;q\right)_{n}}{\widehat{R}}_{n}\!\left(\mu(x)\right)}}}$

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

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

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

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

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

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

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

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

${\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(1-\alpha q^{x+1})(1-\beta\delta q^{x+1})(1-\gamma q^{x+1})(1-\gamma\delta q^{x+1})}{(1-\gamma\delta q^{2x+1})(1-\gamma\delta q^{2x+2})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle y(x)=R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x+1);\alpha,\beta,\gamma,\delta\,|\,q\right)-R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right){}=\frac{q^{-n-x}(1-q^{n})(1-\alpha\beta q^{n+1})(1-\gamma\delta q^{2x+2})}{(1-\alpha q)(1-\beta\delta q)(1-\gamma q)}{}R_{n-1}\!\left(\mu(x);\alpha q,\beta q,\gamma q,\delta\,|\,q\right)}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle\frac{\Delta R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)}{\Delta\mu(x)}{}=\frac{q^{-n+1}(1-q^{n})(1-\alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-\beta\delta q)(1-\gamma q)}{}R_{n-1}\!\left(\mu(x);\alpha q,\beta q,\gamma q,\delta\,|\,q\right)}}}$

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

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

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

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle(1-\alpha q^{x})(1-\beta\delta q^{x})(1-\gamma q^{x})(1-\gamma\delta q^{x})R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right){}-(1-q^{x})(1-\delta q^{x})(\alpha-\gamma\delta q^{x})(\beta-\gamma q^{x})R_{n}\!\left(\mu(x-1);\alpha,\beta,\gamma,\delta\,|\,q\right){}=q^{x}(1-\alpha)(1-\beta\delta)(1-\gamma)(1-\gamma\delta q^{2x}){}R_{n+1}\!\left(\mu(x);\alpha q^{-1},\beta q^{-1},\gamma q^{-1},\delta\,|\,q\right)}}}$

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

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

${\displaystyle{\displaystyle{\displaystyle\nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)}}}}$

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

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

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

Generating functions

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{-x},\alpha\gamma^{-1}\delta^{-1}q^{-x}}{\alpha q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\beta\delta q^{x+1},\gamma q^{x+1}}{\beta q}{q}{q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(\beta\delta q,\gamma q;q\right)_{n}}{\left(\beta q,q;q\right)_{n}}R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)t^{n}{}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\beta\delta q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N}}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{-x},\beta\gamma^{-1}q^{-x}}{\beta\delta q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\alpha q^{x+1},\gamma q^{x+1}}{\alpha\delta^{-1}q}{q}{q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(\alpha q,\gamma q;q\right)_{n}}{\left(\alpha\delta^{-1}q,q;q\right)_{n}}R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)t^{n}{}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\alpha q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N}}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{-x},\delta^{-1}q^{-x}}{\gamma q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\alpha q^{x+1},\beta\delta q^{x+1}}{\alpha\beta\gamma^{-1}q}{q}{q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(\alpha q,\beta\delta q;q\right)_{n}}{\left(\alpha\beta\gamma^{-1}q,q;q\right)_{n}}R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)t^{n}{}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\alpha q=q^{-N}\quad\textrm{or}\quad\beta\delta q=q^{-N}}}}$

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

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

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

Limit relations

q-Racah polynomial to Big q-Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);a,b,c,0\,|\,q\right)=P_{n}\!\left(q^{-x};a,b,c;q\right)}}}$

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

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

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

q-Racah polynomial to q-Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,q^{-N-1},0\,|\,q\right)=Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right)}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,0,\beta^{-1}q^{-N-1}\,|\,q\right)=Q_{n}\!\left(q^{-x};\alpha,\beta,N;q\right)}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);q^{-N-1},\beta\gamma q^{N+1},\gamma,0\,|\,q\right)=Q_{n}\!\left(q^{-x};\gamma,\beta,N;q\right)}}}$

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

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

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

q-Racah polynomial to Dual q-Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);q^{-N-1},0,\gamma,\delta\,|\,q\right)=R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);0,\delta^{-1}q^{-N-1},\gamma,\delta\,|\,q\right)=R_{n}\!\left(\mu(x);\gamma,\delta,N\right){q}}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,0,q^{-N-1},\alpha\delta q^{N+1}\,|\,q\right)=R_{n}\!\left({\tilde{\mu}}(x);\alpha,\delta,N\right){q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\tilde{\mu}}(x)=q^{-x}+\alpha\delta q^{x+1}}}}$ &

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

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

q-Racah polynomial to q-Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(q^{-x};q^{-N-1},-pq^{N},0,0\,|\,q\right)=K_{n}\!\left(q^{-x};p,N;q\right)}}}$

q-Racah polynomial to Dual q-Krawtchouk polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);0,0,q^{-N-1},c\,|\,q\right)=K_{n}\!\left(\lambda(x);c,N|q\right)}}}$

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

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

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

q-Racah polynomial to Racah polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}R_{n}\!\left(\mu(x);q^{\alpha},q^{\beta},q^{\gamma},q^{\delta}\,|\,q\right)=R_{n}\!\left(\lambda(x);\alpha,\beta,\gamma,\delta\right)}}}$

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

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

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

Remarks

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(2a\cos\theta;abq^{-1},cdq^{-1},adq^{-1},ad^{-1}\,|\,q\right)=\frac{a^{n}p_{n}\!\left(x;a,b,c,d\,|\,q\right)}{\left(ab,ac,ad;q\right)_{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q^{-1}\right)=R_{n}\!\left({\tilde{\mu}}(x);\alpha^{-1},\beta^{-1},\gamma^{-1},\delta^{-1}\,|\,q\right)}}}$

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

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

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

Koornwinder Addendum: q-Racah

Symmetry

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(x;\alpha,\beta,q^{-N-1},\delta\,|\,q\right)=\frac{\left(\beta q,\alpha\delta^{-1}q;q\right)_{n}}{\left(\alpha q,\beta\delta q;q\right)_{n}}\delta^{n}R_{n}\!\left(\delta^{-1}x;\beta,\alpha,q^{-N-1},\delta^{-1}\,|\,q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(x;\alpha,\beta,q^{-N-1},-1\,|\,q\right)=\frac{\left(\beta q,-\alpha q;q\right)_{n}}{\left(\alpha q,-\beta q;q\right)_{n}}(-1)^{n}R_{n}\!\left(-x;\beta,\alpha,q^{-N-1},-1\,|\,q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(x;\alpha,\alpha,q^{-N-1},-1\,|\,q\right)=(-1)^{n}R_{n}\!\left(-x;\alpha,\alpha,q^{-N-1},-1\,|\,q\right)}}}$

Trivial symmetry

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)=R_{n}\!\left(x;\beta\delta,\alpha\delta^{-1},\gamma,\delta\,|\,q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)=R_{n}\!\left(x;\gamma,\alpha\beta\gamma^{-1},\alpha,\gamma\delta\alpha^{-1}\,|\,q\right)}}}$

Duality

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(q^{-y}+\gamma\delta q^{y+1};q^{-N-1},\beta,\gamma,\delta\,|\,q\right)=R_{y}\!\left(q^{-n}+\beta q^{n-N};\gamma,\delta,q^{-N-1},\beta\,|\,q\right)(n,y}}}$
${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(q^{-y}+\gamma\delta q^{y+1};q^{-N-1},\beta,\gamma,\delta\,|\,q\right)=0,1,\ldots,N)}}}$

<< Askey-Wilson

q-Racah

Continuous dual q-Hahn >>