\qRacah 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 \qRacah 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\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}}}} {\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} }
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}}} &
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 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) }
∑ x = 0 N \qPochhammer α q , β δ q , γ q , γ δ q q x \qPochhammer 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 ) \qRacah m @ @ μ ( x ) α β γ δ q \qRacah n @ @ μ ( x ) α β γ δ q = h n \Kronecker m n superscript subscript 𝑥 0 𝑁 \qPochhammer 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝛾 𝛿 𝑞 𝑞 𝑥 \qPochhammer 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 \qRacah 𝑚 @ @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 \qRacah 𝑛 @ @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript ℎ 𝑛 \Kronecker 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\qPochhammer{% \alpha q,\beta\delta q,\gamma q,\gamma\delta q}{q}{x}}{\qPochhammer{q,\alpha^{% -1}\gamma\delta q,\beta^{-1}\gamma q,\delta q}{q}{x}}{}\frac{(1-\gamma\delta q% ^{2x+1})}{(\alpha\beta q)^{x}(1-\gamma\delta q)}\qRacah{m}@@{\mu(x)}{\alpha}{% \beta}{\gamma}{\delta}{q}\qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q% }=h_{n}\,\Kronecker{m}{n}}}} {\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} }
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}}} &
\qRacah n @ @ μ ( x ) α β γ δ q := \qRacah n @ μ ( x ) α β γ δ q assign \qRacah 𝑛 @ @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 \qRacah 𝑛 @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle\qRacah{n}@@{\mu(x)}{\alpha}{\beta}{% \gamma}{\delta}{q}:=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}}}} {\displaystyle \qRacah{n}@@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}:=\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} }
- ( 1 - q - x ) ( 1 - γ δ q x + 1 ) \qRacah n @ @ μ ( x ) α β γ δ q = A n \qRacah n + 1 @ @ μ ( x ) α β γ δ q - ( A n + C n ) \qRacah n @ @ μ ( x ) α β γ δ q + C n \qRacah n - 1 @ @ μ ( x ) α β γ δ q 1 superscript 𝑞 𝑥 1 𝛾 𝛿 superscript 𝑞 𝑥 1 \qRacah 𝑛 @ @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 \qRacah 𝑛 1 @ @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 \qRacah 𝑛 @ @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐶 𝑛 \qRacah 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)\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}}}} {\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} }
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}}} &
x \monicqRacah n @ @ x α β γ δ q = \monicqRacah n + 1 @ @ x α β γ δ q + [ 1 + γ δ q - ( A n + C n ) ] \monicqRacah n @ @ x α β γ δ q + A n - 1 C n \monicqRacah n - 1 @ @ x α β γ δ q 𝑥 \monicqRacah 𝑛 @ @ 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 \monicqRacah 𝑛 1 @ @ 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 delimited-[] 1 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 \monicqRacah 𝑛 @ @ 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 \monicqRacah 𝑛 1 @ @ 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle x\monicqRacah{n}@@{x}{\alpha}{\beta% }{\gamma}{\delta}{q}=\monicqRacah{n+1}@@{x}{\alpha}{\beta}{\gamma}{\delta}{q}+% \left[1+\gamma\delta q-(A_{n}+C_{n})\right]\monicqRacah{n}@@{x}{\alpha}{\beta}% {\gamma}{\delta}{q}+A_{n-1}C_{n}\monicqRacah{n-1}@@{x}{\alpha}{\beta}{\gamma}{% \delta}{q}}}} {\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} }
\qRacah n @ μ ( x ) α β γ δ q = \qPochhammer α β q n + 1 q n \qPochhammer α q , β δ q , γ q q n \monicqRacah n @ @ μ ( x ) α β γ δ q \qRacah 𝑛 @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 \qPochhammer 𝛼 𝛽 superscript 𝑞 𝑛 1 𝑞 𝑛 \qPochhammer 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑛 \monicqRacah 𝑛 @ @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle\qRacah{n}@{\mu(x)}{\alpha}{\beta}{% \gamma}{\delta}{q}=\frac{\qPochhammer{\alpha\beta q^{n+1}}{q}{n}}{\qPochhammer% {\alpha q,\beta\delta q,\gamma q}{q}{n}}\monicqRacah{n}@@{\mu(x)}{\alpha}{% \beta}{\gamma}{\delta}{q}}}} {\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} }
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 }
w ( x ) := w ( x ; α , β , γ , δ | q ) = \qPochhammer α q , β δ q , γ q , γ δ q q x \qPochhammer 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 \qPochhammer 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝛾 𝛿 𝑞 𝑞 𝑥 \qPochhammer 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{\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)}}}} & y ( x ) = \qRacah n @ μ ( x ) α β γ δ q 𝑦 𝑥 \qRacah 𝑛 @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=\qRacah{n}@{\mu(x)}{\alpha}{% \beta}{\gamma}{\delta}{q}}}} & 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}}} &
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) }
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 ) = \qRacah n @ μ ( x ) α β γ δ q 𝑦 𝑥 \qRacah 𝑛 @ 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=\qRacah{n}@{\mu(x)}{\alpha}{% \beta}{\gamma}{\delta}{q}}}} & 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}}} &
\qRacah n @ μ ( x + 1 ) α β γ δ q - \qRacah 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 ) \qRacah n - 1 @ μ ( x ) α q β q γ q δ q \qRacah 𝑛 @ 𝜇 𝑥 1 𝛼 𝛽 𝛾 𝛿 𝑞 \qRacah 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 \qRacah 𝑛 1 @ 𝜇 𝑥 𝛼 𝑞 𝛽 𝑞 𝛾 𝑞 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle\qRacah{n}@{\mu(x+1)}{\alpha}{\beta}% {\gamma}{\delta}{q}-\qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}{}=% \frac{q^{-n-x}(1-q^{n})(1-\alpha\beta q^{n+1})(1-\gamma\delta q^{2x+2})}{(1-% \alpha q)(1-\beta\delta q)(1-\gamma q)}{}\qRacah{n-1}@{\mu(x)}{\alpha q}{\beta q% }{\gamma q}{\delta}{q}}}} {\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} }
Δ \qRacah 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 ) \qRacah n - 1 @ μ ( x ) α q β q γ q δ q Δ \qRacah 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 \qRacah 𝑛 1 @ 𝜇 𝑥 𝛼 𝑞 𝛽 𝑞 𝛾 𝑞 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\Delta\qRacah{n}@{\mu(x)}{% \alpha}{\beta}{\gamma}{\delta}{q}}{\Delta\mu(x)}{}=\frac{q^{-n+1}(1-q^{n})(1-% \alpha\beta q^{n+1})}{(1-q)(1-\alpha q)(1-\beta\delta q)(1-\gamma q)}{}\qRacah% {n-1}@{\mu(x)}{\alpha q}{\beta q}{\gamma q}{\delta}{q}}}} {\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} }
( 1 - α q x ) ( 1 - β δ q x ) ( 1 - γ q x ) ( 1 - γ δ q x ) \qRacah n @ μ ( x ) α β γ δ q - ( 1 - q x ) ( 1 - δ q x ) ( α - γ δ q x ) ( β - γ q x ) \qRacah n @ μ ( x - 1 ) α β γ δ q = q x ( 1 - α ) ( 1 - β δ ) ( 1 - γ ) ( 1 - γ δ q 2 x ) \qRacah 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 \qRacah 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 \qRacah 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 \qRacah 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})\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}}}} {\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} }
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