Formula:KLS:14.02:12

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x ⁢ R ^ n ⁡ ( x ) = R ^ n + 1 ⁡ ( x ) + [ 1 + γ ⁢ δ ⁢ q - ( A n + C n ) ] ⁢ R ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ R ^ n - 1 ⁡ ( x ) 𝑥 q-Racah-polynomial-monic-R 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 q-Racah-polynomial-monic-R 𝑛 1 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 delimited-[] 1 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Racah-polynomial-monic-R 𝑛 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 q-Racah-polynomial-monic-R 𝑛 1 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{R}}_{n}\!\left(x\right)=% {\widehat{R}}_{n+1}\!\left(x\right)+\left[1+\gamma\delta q-(A_{n}+C_{n})\right% ]{\widehat{R}}_{n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{R}}_{n-1}\!\left(x% \right)}}}

Substitution(s)

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})}}}}


Proof

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Symbols List

& : logical and
R ^ n subscript ^ 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{R}}_{n}}}}  : monic q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Racah polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqRacah

Bibliography

Equation in Section 14.2 of KLS.

URL links

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