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Latest revision as of 00:33, 6 March 2017

q-Racah

Basic hypergeometric representation

R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) = \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , α ⁢ β ⁢ q n + 1 , q - x , γ ⁢ δ ⁢ q x + 1 ⁢ α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q ⁢ q ⁢ q q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 𝛼 𝛽 superscript 𝑞 𝑛 1 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑞 {\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}}}} {\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): n = 0 , 1 , 2 , … , N 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}


Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


α ⁢ q = q - N   or   β ⁢ δ ⁢ q = q - N   or   γ ⁢ q = q - N formulae-sequence 𝛼 𝑞 superscript 𝑞 𝑁 or formulae-sequence 𝛽 𝛿 𝑞 superscript 𝑞 𝑁 or 𝛾 𝑞 superscript 𝑞 𝑁 {\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 \alpha q=q^{-N}\quad\textrm{or}\quad\beta\delta q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N} }
( q - x , γ ⁢ δ ⁢ q x + 1 ; q ) k = ∏ j = 0 k - 1 ( 1 - μ ⁢ ( x ) ⁢ q j + γ ⁢ δ ⁢ q 2 ⁢ j + 1 ) q-Pochhammer-symbol superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝑞 𝑘 superscript subscript product 𝑗 0 𝑘 1 1 𝜇 𝑥 superscript 𝑞 𝑗 𝛾 𝛿 superscript 𝑞 2 𝑗 1 {\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)}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Orthogonality relation(s)

∑ x = 0 N ( α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q , γ ⁢ δ ⁢ q ; q ) x ( q , α - 1 ⁢ γ ⁢ δ ⁢ q , β - 1 ⁢ γ ⁢ q , δ ⁢ q ; q ) x ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 1 ) ( α ⁢ β ⁢ q ) x ⁢ ( 1 - γ ⁢ δ ⁢ q ) ⁢ R m ⁡ ( μ ⁢ ( x ) ) ⁢ R n ⁡ ( μ ⁢ ( x ) ) = h n ⁢ δ m , n superscript subscript 𝑥 0 𝑁 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝛾 𝛿 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝛼 1 𝛾 𝛿 𝑞 superscript 𝛽 1 𝛾 𝑞 𝛿 𝑞 𝑞 𝑥 1 𝛾 𝛿 superscript 𝑞 2 𝑥 1 superscript 𝛼 𝛽 𝑞 𝑥 1 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑚 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript ℎ 𝑛 Kronecker-delta 𝑚 𝑛 {\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^{2% x+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}}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
h n = ( α - 1 ⁢ β - 1 ⁢ γ , α - 1 ⁢ δ , β - 1 , γ ⁢ δ ⁢ q 2 ; q ) ∞ ( α - 1 ⁢ β - 1 ⁢ q - 1 , α - 1 ⁢ γ ⁢ δ ⁢ q , β - 1 ⁢ γ ⁢ q , δ ⁢ q ; q ) ∞ ⁢ ( 1 - α ⁢ β ⁢ q ) ⁢ ( γ ⁢ δ ⁢ q ) n ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( q , α ⁢ β ⁢ γ - 1 ⁢ q , α ⁢ δ - 1 ⁢ q , β ⁢ q ; q ) n ( α ⁢ q , α ⁢ β ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q ; q ) n = { ( β - 1 , γ ⁢ δ ⁢ q 2 ; q ) N ( β - 1 ⁢ γ ⁢ q , δ ⁢ q ; q ) N ⁢ ( 1 - β ⁢ q - N ) ⁢ ( γ ⁢ δ ⁢ q ) n ( 1 - β ⁢ q 2 ⁢ n - N ) ⁢ ( q , β ⁢ q , β ⁢ γ - 1 ⁢ q - N , δ - 1 ⁢ q - N ; q ) n ( β ⁢ q - N , β ⁢ δ ⁢ q , γ ⁢ q , q - N ; q ) n < b r / >   if   α q = q - N ( α ⁢ β ⁢ q 2 , β ⁢ γ - 1 ; q ) N ( α ⁢ β ⁢ γ - 1 ⁢ q , β ⁢ q ; q ) N ⁢ ( 1 - α ⁢ β ⁢ q ) ⁢ ( β - 1 ⁢ γ ⁢ q - N ) n ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( q , α ⁢ β ⁢ q N + 2 , α ⁢ β ⁢ γ - 1 ⁢ q , β ⁢ q ; q ) n ( α ⁢ q , α ⁢ β ⁢ q , γ ⁢ q , q - N ; q ) n < b r / >   if   β δ q = q - N ( α ⁢ β ⁢ q 2 , δ - 1 ; q ) N ( α ⁢ δ - 1 ⁢ q , β ⁢ q ; q ) N ⁢ ( 1 - α ⁢ β ⁢ q ) ⁢ ( δ ⁢ q - N ) n ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( q , α ⁢ β ⁢ q N + 2 , α ⁢ δ - 1 ⁢ q , β ⁢ q ; q ) n ( α ⁢ q , α ⁢ β ⁢ q , β ⁢ δ ⁢ q , q - N ; q ) n < b r / >   if   γ q = q - N subscript ℎ 𝑛 q-Pochhammer-symbol superscript 𝛼 1 superscript 𝛽 1 𝛾 superscript 𝛼 1 𝛿 superscript 𝛽 1 𝛾 𝛿 superscript 𝑞 2 𝑞 q-Pochhammer-symbol superscript 𝛼 1 superscript 𝛽 1 superscript 𝑞 1 superscript 𝛼 1 𝛾 𝛿 𝑞 superscript 𝛽 1 𝛾 𝑞 𝛿 𝑞 𝑞 1 𝛼 𝛽 𝑞 superscript 𝛾 𝛿 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝛼 𝛽 superscript 𝛾 1 𝑞 𝛼 superscript 𝛿 1 𝑞 𝛽 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛼 𝛽 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑛 cases q-Pochhammer-symbol superscript 𝛽 1 𝛾 𝛿 superscript 𝑞 2 𝑞 𝑁 q-Pochhammer-symbol superscript 𝛽 1 𝛾 𝑞 𝛿 𝑞 𝑞 𝑁 1 𝛽 superscript 𝑞 𝑁 superscript 𝛾 𝛿 𝑞 𝑛 1 𝛽 superscript 𝑞 2 𝑛 𝑁 q-Pochhammer-symbol 𝑞 𝛽 𝑞 𝛽 superscript 𝛾 1 superscript 𝑞 𝑁 superscript 𝛿 1 superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝛽 superscript 𝑞 𝑁 𝛽 𝛿 𝑞 𝛾 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 fragments b r italic-  if italic-  α q superscript 𝑞 𝑁 missing-subexpression missing-subexpression q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 2 𝛽 superscript 𝛾 1 𝑞 𝑁 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝛾 1 𝑞 𝛽 𝑞 𝑞 𝑁 1 𝛼 𝛽 𝑞 superscript superscript 𝛽 1 𝛾 superscript 𝑞 𝑁 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝛼 𝛽 superscript 𝑞 𝑁 2 𝛼 𝛽 superscript 𝛾 1 𝑞 𝛽 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛼 𝛽 𝑞 𝛾 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 fragments b r italic-  if italic-  β δ q superscript 𝑞 𝑁 missing-subexpression missing-subexpression q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 2 superscript 𝛿 1 𝑞 𝑁 q-Pochhammer-symbol 𝛼 superscript 𝛿 1 𝑞 𝛽 𝑞 𝑞 𝑁 1 𝛼 𝛽 𝑞 superscript 𝛿 superscript 𝑞 𝑁 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝛼 𝛽 superscript 𝑞 𝑁 2 𝛼 superscript 𝛿 1 𝑞 𝛽 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛼 𝛽 𝑞 𝛽 𝛿 𝑞 superscript 𝑞 𝑁 𝑞 𝑛 fragments b r italic-  if italic-  γ q superscript 𝑞 𝑁 {\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.}}} } &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


R n ⁡ ( μ ⁢ ( x ) ) := R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) assign q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x)\right):=R_{n}\!% \left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)}}} {\displaystyle \qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}:=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} }

Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Recurrence relation

- ( 1 - q - x ) ⁢ ( 1 - γ ⁢ δ ⁢ q x + 1 ) ⁢ R n ⁡ ( μ ⁢ ( x ) ) = A n ⁢ R n + 1 ⁡ ( μ ⁢ ( x ) ) - ( A n + C n ) ⁢ R n ⁡ ( μ ⁢ ( x ) ) + C n ⁢ R n - 1 ⁡ ( μ ⁢ ( x ) ) 1 superscript 𝑞 𝑥 1 𝛾 𝛿 superscript 𝑞 𝑥 1 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 q-Racah-polynomial-R 𝑛 1 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐶 𝑛 q-Racah-polynomial-R 𝑛 1 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\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)}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
C n = q ⁢ ( 1 - q n ) ⁢ ( 1 - β ⁢ q n ) ⁢ ( γ - α ⁢ β ⁢ q n ) ⁢ ( δ - α ⁢ q n ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝑞 1 superscript 𝑞 𝑛 1 𝛽 superscript 𝑞 𝑛 𝛾 𝛼 𝛽 superscript 𝑞 𝑛 𝛿 𝛼 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 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})}}}} &
A n = ( 1 - α ⁢ q n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ⁢ ( 1 - β ⁢ δ ⁢ q n + 1 ) ⁢ ( 1 - γ ⁢ q n + 1 ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 1 𝛼 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝛽 𝛿 superscript 𝑞 𝑛 1 1 𝛾 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 2 {\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})}}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Monic recurrence relation

x ⁢ R ^ n ⁡ ( x ) = R ^ n + 1 ⁡ ( x ) + [ 1 + γ ⁢ δ ⁢ q - ( A n + C n ) ] ⁢ R ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ R ^ n - 1 ⁡ ( x ) 𝑥 q-Racah-polynomial-monic-R 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-monic-R 𝑛 1 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 delimited-[] 1 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Racah-polynomial-monic-R 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 q-Racah-polynomial-monic-R 𝑛 1 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\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)}}} {\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): C n = q ⁢ ( 1 - q n ) ⁢ ( 1 - β ⁢ q n ) ⁢ ( γ - α ⁢ β ⁢ q n ) ⁢ ( δ - α ⁢ q n ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝑞 1 superscript 𝑞 𝑛 1 𝛽 superscript 𝑞 𝑛 𝛾 𝛼 𝛽 superscript 𝑞 𝑛 𝛿 𝛼 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 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})}}}} &
A n = ( 1 - α ⁢ q n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ⁢ ( 1 - β ⁢ δ ⁢ q n + 1 ) ⁢ ( 1 - γ ⁢ q n + 1 ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 1 𝛼 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝛽 𝛿 superscript 𝑞 𝑛 1 1 𝛾 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 2 {\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})}}}}


R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) = ( α ⁢ β ⁢ q n + 1 ; q ) n ( α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q ; q ) n ⁢ R ^ n ⁡ ( μ ⁢ ( x ) ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 𝑛 1 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑛 q-Racah-polynomial-monic-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\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)}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


q-Difference equation

Δ ⁢ [ w ⁢ ( x - 1 ) ⁢ B ⁢ ( x - 1 ) ⁢ Δ ⁢ y ⁢ ( x - 1 ) ] - q - n ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ⁢ w ⁢ ( x ) ⁢ y ⁢ ( x ) = 0 Δ delimited-[] 𝑤 𝑥 1 𝐵 𝑥 1 Δ 𝑦 𝑥 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 1 𝑤 𝑥 𝑦 𝑥 0 {\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}}} {\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): B ⁢ ( x ) = ( 1 - α ⁢ q x + 1 ) ⁢ ( 1 - β ⁢ δ ⁢ q x + 1 ) ⁢ ( 1 - γ ⁢ q x + 1 ) ⁢ ( 1 - γ ⁢ δ ⁢ q x + 1 ) ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 1 ) ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 2 ) 𝐵 𝑥 1 𝛼 superscript 𝑞 𝑥 1 1 𝛽 𝛿 superscript 𝑞 𝑥 1 1 𝛾 superscript 𝑞 𝑥 1 1 𝛾 𝛿 superscript 𝑞 𝑥 1 1 𝛾 𝛿 superscript 𝑞 2 𝑥 1 1 𝛾 𝛿 superscript 𝑞 2 𝑥 2 {\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})}}}} &

w ⁢ ( x ) := w ⁢ ( x ; α , β , γ , δ | q ) = ( α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q , γ ⁢ δ ⁢ q ; q ) x ( q , α - 1 ⁢ γ ⁢ δ ⁢ q , β - 1 ⁢ γ ⁢ q , δ ⁢ q ; q ) x ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 1 ) ( α ⁢ β ⁢ q ) x ⁢ ( 1 - γ ⁢ δ ⁢ q ) assign 𝑤 𝑥 𝑤 𝑥 𝛼 𝛽 𝛾 conditional 𝛿 𝑞 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝛾 𝛿 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝛼 1 𝛾 𝛿 𝑞 superscript 𝛽 1 𝛾 𝑞 𝛿 𝑞 𝑞 𝑥 1 𝛾 𝛿 superscript 𝑞 2 𝑥 1 superscript 𝛼 𝛽 𝑞 𝑥 1 𝛾 𝛿 𝑞 {\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)}}}} &
y ⁢ ( x ) = R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) 𝑦 𝑥 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=R_{n}\!\left(\mu(x);\alpha,% \beta,\gamma,\delta\,|\,q\right)}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &
λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


q - n ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( x + 1 ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( x - 1 ) superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 1 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 1 delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 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)}}} {\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): D ⁢ ( x ) = q ⁢ ( 1 - q x ) ⁢ ( 1 - δ ⁢ q x ) ⁢ ( β - γ ⁢ q x ) ⁢ ( α - γ ⁢ δ ⁢ q x ) ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x ) ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 1 ) 𝐷 𝑥 𝑞 1 superscript 𝑞 𝑥 1 𝛿 superscript 𝑞 𝑥 𝛽 𝛾 superscript 𝑞 𝑥 𝛼 𝛾 𝛿 superscript 𝑞 𝑥 1 𝛾 𝛿 superscript 𝑞 2 𝑥 1 𝛾 𝛿 superscript 𝑞 2 𝑥 1 {\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})}}}} &

B ⁢ ( x ) = ( 1 - α ⁢ q x + 1 ) ⁢ ( 1 - β ⁢ δ ⁢ q x + 1 ) ⁢ ( 1 - γ ⁢ q x + 1 ) ⁢ ( 1 - γ ⁢ δ ⁢ q x + 1 ) ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 1 ) ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 2 ) 𝐵 𝑥 1 𝛼 superscript 𝑞 𝑥 1 1 𝛽 𝛿 superscript 𝑞 𝑥 1 1 𝛾 superscript 𝑞 𝑥 1 1 𝛾 𝛿 superscript 𝑞 𝑥 1 1 𝛾 𝛿 superscript 𝑞 2 𝑥 1 1 𝛾 𝛿 superscript 𝑞 2 𝑥 2 {\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})}}}} &
y ⁢ ( x ) = R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) 𝑦 𝑥 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=R_{n}\!\left(\mu(x);\alpha,% \beta,\gamma,\delta\,|\,q\right)}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &
λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Forward shift operator

R n ⁡ ( μ ⁢ ( x + 1 ) ; α , β , γ , δ | q ) - R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) = q - n - x ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 2 ) ( 1 - α ⁢ q ) ⁢ ( 1 - β ⁢ δ ⁢ q ) ⁢ ( 1 - γ ⁢ q ) ⁢ R n - 1 ⁡ ( μ ⁢ ( x ) ; α ⁢ q , β ⁢ q , γ ⁢ q , δ | q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 1 𝛼 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 𝑞 𝑛 𝑥 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝛾 𝛿 superscript 𝑞 2 𝑥 2 1 𝛼 𝑞 1 𝛽 𝛿 𝑞 1 𝛾 𝑞 q-Racah-polynomial-R 𝑛 1 𝜇 𝑥 𝛼 𝑞 𝛽 𝑞 𝛾 𝑞 𝛿 𝑞 {\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)}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Δ ⁢ R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) Δ ⁢ μ ⁢ ( x ) = q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ( 1 - q ) ⁢ ( 1 - α ⁢ q ) ⁢ ( 1 - β ⁢ δ ⁢ q ) ⁢ ( 1 - γ ⁢ q ) ⁢ R n - 1 ⁡ ( μ ⁢ ( x ) ; α ⁢ q , β ⁢ q , γ ⁢ q , δ | q ) Δ q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 Δ 𝜇 𝑥 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝑞 1 𝛼 𝑞 1 𝛽 𝛿 𝑞 1 𝛾 𝑞 q-Racah-polynomial-R 𝑛 1 𝜇 𝑥 𝛼 𝑞 𝛽 𝑞 𝛾 𝑞 𝛿 𝑞 {\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)}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Backward shift operator

( 1 - α ⁢ q x ) ⁢ ( 1 - β ⁢ δ ⁢ q x ) ⁢ ( 1 - γ ⁢ q x ) ⁢ ( 1 - γ ⁢ δ ⁢ q x ) ⁢ R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) - ( 1 - q x ) ⁢ ( 1 - δ ⁢ q x ) ⁢ ( α - γ ⁢ δ ⁢ q x ) ⁢ ( β - γ ⁢ q x ) ⁢ R n ⁡ ( μ ⁢ ( x - 1 ) ; α , β , γ , δ | q ) = q x ⁢ ( 1 - α ) ⁢ ( 1 - β ⁢ δ ) ⁢ ( 1 - γ ) ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x ) ⁢ R n + 1 ⁡ ( μ ⁢ ( x ) ; α ⁢ q - 1 , β ⁢ q - 1 , γ ⁢ q - 1 , δ | q ) 1 𝛼 superscript 𝑞 𝑥 1 𝛽 𝛿 superscript 𝑞 𝑥 1 𝛾 superscript 𝑞 𝑥 1 𝛾 𝛿 superscript 𝑞 𝑥 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 1 superscript 𝑞 𝑥 1 𝛿 superscript 𝑞 𝑥 𝛼 𝛾 𝛿 superscript 𝑞 𝑥 𝛽 𝛾 superscript 𝑞 𝑥 q-Racah-polynomial-R 𝑛 𝜇 𝑥 1 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 𝑞 𝑥 1 𝛼 1 𝛽 𝛿 1 𝛾 1 𝛾 𝛿 superscript 𝑞 2 𝑥 q-Racah-polynomial-R 𝑛 1 𝜇 𝑥 𝛼 superscript 𝑞 1 𝛽 superscript 𝑞 1 𝛾 superscript 𝑞 1 𝛿 𝑞 {\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% )}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


∇ ⁡ [ w ~ ⁢ ( x ; α , β , γ , δ | q ) ⁢ R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) ] ∇ ⁡ μ ⁢ ( x ) = 1 ( 1 - q ) ⁢ ( 1 - γ ⁢ δ ) ⁢ w ~ ⁢ ( x ; α ⁢ q - 1 , β ⁢ q - 1 , γ ⁢ q - 1 , δ | q ) ⁢ R n + 1 ⁡ ( μ ⁢ ( x ) ; α ⁢ q - 1 , β ⁢ q - 1 , γ ⁢ q - 1 , δ | q ) ∇ ~ 𝑤 𝑥 𝛼 𝛽 𝛾 conditional 𝛿 𝑞 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 ∇ 𝜇 𝑥 1 1 𝑞 1 𝛾 𝛿 ~ 𝑤 𝑥 𝛼 superscript 𝑞 1 𝛽 superscript 𝑞 1 𝛾 superscript 𝑞 1 conditional 𝛿 𝑞 q-Racah-polynomial-R 𝑛 1 𝜇 𝑥 𝛼 superscript 𝑞 1 𝛽 superscript 𝑞 1 𝛾 superscript 𝑞 1 𝛿 𝑞 {\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)}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


w ~ ⁢ ( x ; α , β , γ , δ | q ) = ( α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q , γ ⁢ δ ⁢ q ; q ) x ( q , α - 1 ⁢ γ ⁢ δ ⁢ q , β - 1 ⁢ γ ⁢ q , δ ⁢ q ; q ) x ⁢ ( α ⁢ β ) x ~ 𝑤 𝑥 𝛼 𝛽 𝛾 conditional 𝛿 𝑞 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝛾 𝛿 𝑞 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝛼 1 𝛾 𝛿 𝑞 superscript 𝛽 1 𝛾 𝑞 𝛿 𝑞 𝑞 𝑥 superscript 𝛼 𝛽 𝑥 {\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}}}}} {\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

w ~ ⁢ ( x ; α , β , γ , δ | q ) ⁢ R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) = ( 1 - q ) n ⁢ ( γ ⁢ δ ⁢ q ; q ) n ⁢ ( ∇ μ ) n ⁢ [ w ~ ⁢ ( x ; α ⁢ q n , β ⁢ q n , γ ⁢ q n , δ | q ) ] ~ 𝑤 𝑥 𝛼 𝛽 𝛾 conditional 𝛿 𝑞 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 1 𝑞 𝑛 q-Pochhammer-symbol 𝛾 𝛿 𝑞 𝑞 𝑛 superscript subscript ∇ 𝜇 𝑛 delimited-[] ~ 𝑤 𝑥 𝛼 superscript 𝑞 𝑛 𝛽 superscript 𝑞 𝑛 𝛾 superscript 𝑞 𝑛 conditional 𝛿 𝑞 {\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]}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


∇ μ := ∇ ∇ ⁡ μ ⁢ ( x ) assign subscript ∇ 𝜇 ∇ ∇ 𝜇 𝑥 {\displaystyle{\displaystyle{\displaystyle\nabla_{\mu}:=\frac{\nabla}{\nabla% \mu(x)}}}} {\displaystyle \nabla_{\mu}:=\frac{\nabla}{\nabla\mu(x)} }

Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Generating functions

\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - x , α ⁢ γ - 1 ⁢ δ - 1 ⁢ q - x ⁢ α ⁢ q ⁢ q ⁢ γ ⁢ δ ⁢ q x + 1 ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ β ⁢ δ ⁢ q x + 1 , γ ⁢ q x + 1 ⁢ β ⁢ q ⁢ q ⁢ q - x ⁢ t = ∑ n = 0 N ( β ⁢ δ ⁢ q , γ ⁢ q ; q ) n ( β ⁢ q , q ; q ) n ⁢ R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) ⁢ t n \qHyperrphis 21 @ @ superscript 𝑞 𝑥 𝛼 superscript 𝛾 1 superscript 𝛿 1 superscript 𝑞 𝑥 𝛼 𝑞 𝑞 𝛾 𝛿 superscript 𝑞 𝑥 1 𝑡 \qHyperrphis 21 @ @ 𝛽 𝛿 superscript 𝑞 𝑥 1 𝛾 superscript 𝑞 𝑥 1 𝛽 𝑞 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛽 𝑞 𝑞 𝑞 𝑛 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 𝑡 𝑛 {\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}{}}}} {\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): if   β ⁢ δ ⁢ q = q - N   or   γ ⁢ q = q - N formulae-sequence if 𝛽 𝛿 𝑞 superscript 𝑞 𝑁 or 𝛾 𝑞 superscript 𝑞 𝑁 {\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\beta\delta q=q^{-N}% \quad\textrm{or}\quad\gamma q=q^{-N}}}}


Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - x , β ⁢ γ - 1 ⁢ q - x ⁢ β ⁢ δ ⁢ q ⁢ q ⁢ γ ⁢ δ ⁢ q x + 1 ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ α ⁢ q x + 1 , γ ⁢ q x + 1 ⁢ α ⁢ δ - 1 ⁢ q ⁢ q ⁢ q - x ⁢ t = ∑ n = 0 N ( α ⁢ q , γ ⁢ q ; q ) n ( α ⁢ δ - 1 ⁢ q , q ; q ) n ⁢ R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) ⁢ t n \qHyperrphis 21 @ @ superscript 𝑞 𝑥 𝛽 superscript 𝛾 1 superscript 𝑞 𝑥 𝛽 𝛿 𝑞 𝑞 𝛾 𝛿 superscript 𝑞 𝑥 1 𝑡 \qHyperrphis 21 @ @ 𝛼 superscript 𝑞 𝑥 1 𝛾 superscript 𝑞 𝑥 1 𝛼 superscript 𝛿 1 𝑞 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol 𝛼 𝑞 𝛾 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 superscript 𝛿 1 𝑞 𝑞 𝑞 𝑛 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 𝑡 𝑛 {\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}{}% }}} {\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): if   α ⁢ q = q - N   or   γ ⁢ q = q - N formulae-sequence if 𝛼 𝑞 superscript 𝑞 𝑁 or 𝛾 𝑞 superscript 𝑞 𝑁 {\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\alpha q=q^{-N}\quad% \textrm{or}\quad\gamma q=q^{-N}}}}


Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q - x , δ - 1 ⁢ q - x ⁢ γ ⁢ q ⁢ q ⁢ γ ⁢ δ ⁢ q x + 1 ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ α ⁢ q x + 1 , β ⁢ δ ⁢ q x + 1 ⁢ α ⁢ β ⁢ γ - 1 ⁢ q ⁢ q ⁢ q - x ⁢ t = ∑ n = 0 N ( α ⁢ q , β ⁢ δ ⁢ q ; q ) n ( α ⁢ β ⁢ γ - 1 ⁢ q , q ; q ) n ⁢ R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) ⁢ t n \qHyperrphis 21 @ @ superscript 𝑞 𝑥 superscript 𝛿 1 superscript 𝑞 𝑥 𝛾 𝑞 𝑞 𝛾 𝛿 superscript 𝑞 𝑥 1 𝑡 \qHyperrphis 21 @ @ 𝛼 superscript 𝑞 𝑥 1 𝛽 𝛿 superscript 𝑞 𝑥 1 𝛼 𝛽 superscript 𝛾 1 𝑞 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝛾 1 𝑞 𝑞 𝑞 𝑛 q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 𝑡 𝑛 {\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}{}}}} {\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): if   α ⁢ q = q - N   or   β ⁢ δ ⁢ q = q - N formulae-sequence if 𝛼 𝑞 superscript 𝑞 𝑁 or 𝛽 𝛿 𝑞 superscript 𝑞 𝑁 {\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\alpha q=q^{-N}\quad% \textrm{or}\quad\beta\delta q=q^{-N}}}}


Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Limit relations

q-Racah polynomial to Big q-Jacobi polynomial

R n ⁡ ( μ ⁢ ( x ) ; a , b , c , 0 | q ) = P n ⁡ ( q - x ; a , b , c ; q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝑎 𝑏 𝑐 0 𝑞 big-q-Jacobi-polynomial-P 𝑛 superscript 𝑞 𝑥 𝑎 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);a,b,c,0\,|\,q% \right)=P_{n}\!\left(q^{-x};a,b,c;q\right)}}} {\displaystyle \qRacah{n}@{\mu(x)}{a}{b}{c}{0}{q}=\bigqJacobi{n}@{q^{-x}}{a}{b}{c}{q} }

Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


q-Racah polynomial to q-Hahn polynomial

R n ⁡ ( μ ⁢ ( x ) ; α , β , q - N - 1 , 0 | q ) = Q n ⁡ ( q - x ; α , β , N ; q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 superscript 𝑞 𝑁 1 0 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\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)}}} {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{q^{-N-1}}{0}{q}=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }

Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


R n ⁡ ( μ ⁢ ( x ) ; α , β , 0 , β - 1 ⁢ q - N - 1 | q ) = Q n ⁡ ( q - x ; α , β , N ; q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 0 superscript 𝛽 1 superscript 𝑞 𝑁 1 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 {\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)}}} {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{0}{\beta^{-1}q^{-N-1}}{q}=\qHahn{n}@{q^{-x}}{\alpha}{\beta}{N}{q} }

Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


R n ⁡ ( μ ⁢ ( x ) ; q - N - 1 , β ⁢ γ ⁢ q N + 1 , γ , 0 | q ) = Q n ⁡ ( q - x ; γ , β , N ; q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 superscript 𝑞 𝑁 1 𝛽 𝛾 superscript 𝑞 𝑁 1 𝛾 0 𝑞 q-Hahn-polynomial-Q 𝑛 superscript 𝑞 𝑥 𝛾 𝛽 𝑁 𝑞 {\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)}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


q-Racah polynomial to Dual q-Hahn polynomial

R n ⁡ ( μ ⁢ ( x ) ; q - N - 1 , 0 , γ , δ | q ) = R n ⁡ ( μ ⁢ ( x ) ; γ , δ , N ) ⁢ q q-Racah-polynomial-R 𝑛 𝜇 𝑥 superscript 𝑞 𝑁 1 0 𝛾 𝛿 𝑞 dual-q-Hahn-R 𝑛 𝜇 𝑥 𝛾 𝛿 𝑁 𝑞 {\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}}}} {\displaystyle \qRacah{n}@{\mu(x)}{q^{-N-1}}{0}{\gamma}{\delta}{q}=\dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q} }

Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


R n ⁡ ( μ ⁢ ( x ) ; 0 , δ - 1 ⁢ q - N - 1 , γ , δ | q ) = R n ⁡ ( μ ⁢ ( x ) ; γ , δ , N ) ⁢ q q-Racah-polynomial-R 𝑛 𝜇 𝑥 0 superscript 𝛿 1 superscript 𝑞 𝑁 1 𝛾 𝛿 𝑞 dual-q-Hahn-R 𝑛 𝜇 𝑥 𝛾 𝛿 𝑁 𝑞 {\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% }}}} {\displaystyle \qRacah{n}@{\mu(x)}{0}{\delta^{-1}q^{-N-1}}{\gamma}{\delta}{q}=\dualqHahn{n}@{\mu(x)}{\gamma}{\delta}{N}{q} }

Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


R n ⁡ ( μ ⁢ ( x ) ; α , 0 , q - N - 1 , α ⁢ δ ⁢ q N + 1 | q ) = R n ⁡ ( μ ~ ⁢ ( x ) ; α , δ , N ) ⁢ q q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 0 superscript 𝑞 𝑁 1 𝛼 𝛿 superscript 𝑞 𝑁 1 𝑞 dual-q-Hahn-R 𝑛 ~ 𝜇 𝑥 𝛼 𝛿 𝑁 𝑞 {\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}}}} {\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): μ ~ ⁢ ( x ) = q - x + α ⁢ δ ⁢ q x + 1 ~ 𝜇 𝑥 superscript 𝑞 𝑥 𝛼 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle{\tilde{\mu}}(x)=q^{-x}+\alpha\delta q% ^{x+1}}}} &

μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &
λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


q-Racah polynomial to q-Krawtchouk polynomial

R n ⁡ ( q - x ; q - N - 1 , - p ⁢ q N , 0 , 0 | q ) = K n ⁡ ( q - x ; p , N ; q ) q-Racah-polynomial-R 𝑛 superscript 𝑞 𝑥 superscript 𝑞 𝑁 1 𝑝 superscript 𝑞 𝑁 0 0 𝑞 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\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)}}} {\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

R n ⁡ ( μ ⁢ ( x ) ; 0 , 0 , q - N - 1 , c | q ) = K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 0 0 superscript 𝑞 𝑁 1 𝑐 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 {\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)}}} {\displaystyle \qRacah{n}@{\mu(x)}{0}{0}{q^{-N-1}}{c}{q}=\dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q} }

Substitution(s): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


q-Racah polynomial to Racah polynomial

lim q → 1 ⁡ R n ⁡ ( μ ⁢ ( x ) ; q α , q β , q γ , q δ | q ) = R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) subscript → 𝑞 1 q-Racah-polynomial-R 𝑛 𝜇 𝑥 superscript 𝑞 𝛼 superscript 𝑞 𝛽 superscript 𝑞 𝛾 superscript 𝑞 𝛿 𝑞 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 {\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)}}} {\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): μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Remarks

R n ⁡ ( 2 ⁢ a ⁢ cos ⁡ θ ; a ⁢ b ⁢ q - 1 , c ⁢ d ⁢ q - 1 , a ⁢ d ⁢ q - 1 , a ⁢ d - 1 | q ) = a n ⁢ p n ⁡ ( x ; a , b , c , d | q ) ( a ⁢ b , a ⁢ c , a ⁢ d ; q ) n q-Racah-polynomial-R 𝑛 2 𝑎 𝜃 𝑎 𝑏 superscript 𝑞 1 𝑐 𝑑 superscript 𝑞 1 𝑎 𝑑 superscript 𝑞 1 𝑎 superscript 𝑑 1 𝑞 superscript 𝑎 𝑛 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑛 {\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 \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}} }
R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q - 1 ) = R n ⁡ ( μ ~ ⁢ ( x ) ; α - 1 , β - 1 , γ - 1 , δ - 1 | q ) q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 superscript 𝑞 1 q-Racah-polynomial-R 𝑛 ~ 𝜇 𝑥 superscript 𝛼 1 superscript 𝛽 1 superscript 𝛾 1 superscript 𝛿 1 𝑞 {\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)}}} {\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): μ ~ ⁢ ( x ) := q - x + γ - 1 ⁢ δ - 1 ⁢ q x + 1 assign ~ 𝜇 𝑥 superscript 𝑞 𝑥 superscript 𝛾 1 superscript 𝛿 1 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle{\tilde{\mu}}(x):=q^{-x}+\gamma^{-1}% \delta^{-1}q^{x+1}}}} &

μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &
λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}} &
μ ⁢ ( x ) := q - x + γ ⁢ δ ⁢ q x + 1 assign 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}} &
μ ⁢ ( x ) = q - x + γ ⁢ δ ⁢ q x + 1 = λ ⁢ ( x ) = q - x + c ⁢ q x - N = q - x + q x + γ + δ + 1 = 2 ⁢ a ⁢ cos ⁡ θ 𝜇 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 superscript 𝑞 𝑥 1 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 superscript 𝑞 𝑥 superscript 𝑞 𝑥 𝛾 𝛿 1 2 𝑎 𝜃 {\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}}} &

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Koornwinder Addendum: q-Racah

Symmetry

R n ⁡ ( x ; α , β , q - N - 1 , δ | q ) = ( β ⁢ q , α ⁢ δ - 1 ⁢ q ; q ) n ( α ⁢ q , β ⁢ δ ⁢ q ; q ) n ⁢ δ n ⁢ R n ⁡ ( δ - 1 ⁢ x ; β , α , q - N - 1 , δ - 1 | q ) q-Racah-polynomial-R 𝑛 𝑥 𝛼 𝛽 superscript 𝑞 𝑁 1 𝛿 𝑞 q-Pochhammer-symbol 𝛽 𝑞 𝛼 superscript 𝛿 1 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝑞 𝑛 superscript 𝛿 𝑛 q-Racah-polynomial-R 𝑛 superscript 𝛿 1 𝑥 𝛽 𝛼 superscript 𝑞 𝑁 1 superscript 𝛿 1 𝑞 {\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 \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} }
R n ⁡ ( x ; α , β , q - N - 1 , - 1 | q ) = ( β ⁢ q , - α ⁢ q ; q ) n ( α ⁢ q , - β ⁢ q ; q ) n ⁢ ( - 1 ) n ⁢ R n ⁡ ( - x ; β , α , q - N - 1 , - 1 | q ) q-Racah-polynomial-R 𝑛 𝑥 𝛼 𝛽 superscript 𝑞 𝑁 1 1 𝑞 q-Pochhammer-symbol 𝛽 𝑞 𝛼 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝑞 𝑞 𝑛 superscript 1 𝑛 q-Racah-polynomial-R 𝑛 𝑥 𝛽 𝛼 superscript 𝑞 𝑁 1 1 𝑞 {\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 \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} }
R n ⁡ ( x ; α , α , q - N - 1 , - 1 | q ) = ( - 1 ) n ⁢ R n ⁡ ( - x ; α , α , q - N - 1 , - 1 | q ) q-Racah-polynomial-R 𝑛 𝑥 𝛼 𝛼 superscript 𝑞 𝑁 1 1 𝑞 superscript 1 𝑛 q-Racah-polynomial-R 𝑛 𝑥 𝛼 𝛼 superscript 𝑞 𝑁 1 1 𝑞 {\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)}}} {\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

R n ⁡ ( x ; α , β , γ , δ | q ) = R n ⁡ ( x ; β ⁢ δ , α ⁢ δ - 1 , γ , δ | q ) q-Racah-polynomial-R 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑛 𝑥 𝛽 𝛿 𝛼 superscript 𝛿 1 𝛾 𝛿 𝑞 {\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 \qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta }{ q}=\qRacah{n}@{x}{\beta\delta}{\alpha\delta^{-1}}{\gamma}{\delta }{ q} }
R n ⁡ ( x ; α , β , γ , δ | q ) = R n ⁡ ( x ; γ , α ⁢ β ⁢ γ - 1 , α , γ ⁢ δ ⁢ α - 1 | q ) q-Racah-polynomial-R 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑛 𝑥 𝛾 𝛼 𝛽 superscript 𝛾 1 𝛼 𝛾 𝛿 superscript 𝛼 1 𝑞 {\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)}}} {\displaystyle \qRacah{n}@{x}{\alpha}{\beta}{\gamma}{\delta }{ q} =\qRacah{n}@{x}{\gamma}{\alpha\beta\gamma^{-1}}{\alpha}{\gamma\delta\alpha^{-1} }{ q} }

Duality

R n ⁡ ( q - y + γ ⁢ δ ⁢ q y + 1 ; q - N - 1 , β , γ , δ | q ) = R y ⁡ ( q - n + β ⁢ q n - N ; γ , δ , q - N - 1 , β | q ) ( n , y fragments q-Racah-polynomial-R 𝑛 superscript 𝑞 𝑦 𝛾 𝛿 superscript 𝑞 𝑦 1 superscript 𝑞 𝑁 1 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-R 𝑦 superscript 𝑞 𝑛 𝛽 superscript 𝑞 𝑛 𝑁 𝛾 𝛿 superscript 𝑞 𝑁 1 𝛽 𝑞 fragments ( n , y {\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 \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 }
R n ⁡ ( q - y + γ ⁢ δ ⁢ q y + 1 ; q - N - 1 , β , γ , δ | q ) = 0 , 1 , … , N ) fragments q-Racah-polynomial-R 𝑛 superscript 𝑞 𝑦 𝛾 𝛿 superscript 𝑞 𝑦 1 superscript 𝑞 𝑁 1 𝛽 𝛾 𝛿 𝑞 0 , 1 , … , N ) {\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)}}} {\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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