Formula:KLS:14.02:09

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- ( 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)}}}

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)}}}


Proof

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Symbols List

& : logical and
R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Racah polynomial : http://dlmf.nist.gov/18.28#E19
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.2 of KLS.

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