Formula:KLS:14.10:45

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x ⁢ C ^ n ⁡ ( x ) = C ^ n + 1 ⁡ ( x ) + ( 1 - q n ) ⁢ ( 1 - β 2 ⁢ q n - 1 ) 4 ⁢ ( 1 - β ⁢ q n - 1 ) ⁢ ( 1 - β ⁢ q n ) ⁢ C ^ n - 1 ⁡ ( x ) 𝑥 continuous-q-ultraspherical-Rogers-polynomial-monic-p 𝑛 𝑥 𝛽 𝑞 continuous-q-ultraspherical-Rogers-polynomial-monic-p 𝑛 1 𝑥 𝛽 𝑞 1 superscript 𝑞 𝑛 1 superscript 𝛽 2 superscript 𝑞 𝑛 1 4 1 𝛽 superscript 𝑞 𝑛 1 1 𝛽 superscript 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial-monic-p 𝑛 1 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{C}}_{n}\!\left(x\right)=% {\widehat{C}}_{n+1}\!\left(x\right)+\frac{(1-q^{n})(1-\beta^{2}q^{n-1})}{4(1-% \beta q^{n-1})(1-\beta q^{n})}{\widehat{C}}_{n-1}\!\left(x\right)}}}

Proof

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Symbols List

C ^ n subscript ^ 𝐶 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{C}}_{n}}}}  : monic continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -ultraspherical/Rogers polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsqUltra

Bibliography

Equation in Section 14.10 of KLS.

URL links

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