Formula:KLS:14.02:20: Difference between revisions

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Latest revision as of 08:36, 22 December 2019


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)}}}

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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