Formula:KLS:14.02:14

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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}}}

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)}}}


Proof

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Symbols List

& : logical and
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
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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