Formula:KLS:14.10:81

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( β + 1 ) ⁢ C n ⁡ [ q - 1 2 ⁢ z ; β | q ] = ( q - 1 2 ⁢ n + q 1 2 ⁢ n ⁢ β ) ⁢ C n ⁡ [ z ; β | q ] + q - 1 2 ⁢ n ⁢ ( β - 1 ) ⁢ ( z - 1 - β ⁢ z ) ⁢ C n - 1 ⁡ [ z ; q ⁢ β | q ] 𝛽 1 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 superscript 𝑞 1 2 𝑧 𝛽 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 2 𝑛 𝛽 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 𝑧 𝛽 𝑞 superscript 𝑞 1 2 𝑛 𝛽 1 superscript 𝑧 1 𝛽 𝑧 continuous-q-ultraspherical-Rogers-polynomial-exponential-argument 𝑛 1 𝑧 𝑞 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle(\beta+1)C_{n}\!\left[q^{-\frac{1}{2% }}z;\beta\,|\,q\right]=(q^{-\frac{1}{2}n}+q^{\frac{1}{2}n}\beta)C_{n}\!\left[z% ;\beta\,|\,q\right]+q^{-\frac{1}{2}n}(\beta-1)(z^{-1}-\beta z)C_{n-1}\!\left[z% ;q\beta\,|\,q\right]}}}

Proof

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Symbols List

C n subscript 𝐶 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -ultraspherical/Rogers polynomial exponential argument : http://drmf.wmflabs.org/wiki/Definition:ctsqUltrae

Bibliography

Equation in Section 14.10 of KLS.

URL links

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