Formula:KLS:14.10:12

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x ⁢ P ^ n ( α , β ) ⁡ ( x ) = P ^ n + 1 ( α , β ) ⁡ ( x ) + 1 2 ⁢ [ q 1 2 ⁢ α + 1 4 + q - 1 2 ⁢ α - 1 4 - ( A n + C n ) ] ⁢ P ^ n ( α , β ) ⁡ ( x ) + 1 4 ⁢ A n - 1 ⁢ C n ⁢ P ^ n - 1 ( α , β ) ⁡ ( x ) 𝑥 continuous-q-Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑞 continuous-q-Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 1 𝑥 𝑞 1 2 delimited-[] superscript 𝑞 1 2 𝛼 1 4 superscript 𝑞 1 2 𝛼 1 4 subscript 𝐴 𝑛 subscript 𝐶 𝑛 continuous-q-Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑞 1 4 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 continuous-q-Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}^{(\alpha,\beta)}_{n}% \!\left(x\right)={\widehat{P}}^{(\alpha,\beta)}_{n+1}\!\left(x\right)+\frac{1}% {2}\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+q^{-\frac{1}{2}\alpha-\frac{1}{4}}-% (A_{n}+C_{n})\right]{\widehat{P}}^{(\alpha,\beta)}_{n}\!\left(x\right){}+\frac% {1}{4}A_{n-1}C_{n}{\widehat{P}}^{(\alpha,\beta)}_{n-1}\!\left(x\right)}}}

Substitution(s)

C n = q 1 2 ⁢ α + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - q n + β ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β ) ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 1 ) ) ( 1 - q 2 ⁢ n + α + β ) ⁢ ( 1 - q 2 ⁢ n + α + β + 1 ) subscript 𝐶 𝑛 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝛽 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{q^{\frac{1}{2}\alpha+% \frac{1}{4}}(1-q^{n})(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n% +\frac{1}{2}(\alpha+\beta+1)})}{(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1% })}}}} &
A n = ( 1 - q n + α + 1 ) ⁢ ( 1 - q n + α + β + 1 ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 1 ) ) ⁢ ( 1 + q n + 1 2 ⁢ ( α + β + 2 ) ) q 1 2 ⁢ α + 1 4 ⁢ ( 1 - q 2 ⁢ n + α + β + 1 ) ⁢ ( 1 - q 2 ⁢ n + α + β + 2 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝛼 1 1 superscript 𝑞 𝑛 𝛼 𝛽 1 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 1 1 superscript 𝑞 𝑛 1 2 𝛼 𝛽 2 superscript 𝑞 1 2 𝛼 1 4 1 superscript 𝑞 2 𝑛 𝛼 𝛽 1 1 superscript 𝑞 2 𝑛 𝛼 𝛽 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+\alpha+1})(1-q^% {n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(% \alpha+\beta+2)})}{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(% 1-q^{2n+\alpha+\beta+2})}}}}


Proof

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Symbols List

& : logical and
P ^ n ( α , β ) subscript superscript ^ 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{P}}^{(\alpha,\beta)}_{n}}}}  : monic continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi

Bibliography

Equation in Section 14.10 of KLS.

URL links

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