Formula:KLS:14.02:23

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∇ ⁡ [ 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)}}}

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)}}}


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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