Formula:KLS:14.10:19

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D q ⁢ P n ( α , β ) ⁡ ( x | q ) = 2 ⁢ q - n + 1 2 ⁢ α + 5 4 ⁢ ( 1 - q n + α + β + 1 ) ( 1 - q ) ⁢ ( 1 + q 1 2 ⁢ ( α + β + 1 ) ) ⁢ ( 1 + q 1 2 ⁢ ( α + β + 2 ) ) ⁢ P n - 1 ( α + 1 , β + 1 ) ⁢ ( x | q ) subscript 𝐷 𝑞 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 2 superscript 𝑞 𝑛 1 2 𝛼 5 4 1 superscript 𝑞 𝑛 𝛼 𝛽 1 1 𝑞 1 superscript 𝑞 1 2 𝛼 𝛽 1 1 superscript 𝑞 1 2 𝛼 𝛽 2 superscript subscript 𝑃 𝑛 1 𝛼 1 𝛽 1 conditional 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}P^{(\alpha,\beta)}_{n}\!\left(% x|q\right)=\frac{2q^{-n+\frac{1}{2}\alpha+\frac{5}{4}}(1-q^{n+\alpha+\beta+1})% }{(1-q)(1+q^{\frac{1}{2}(\alpha+\beta+1)})(1+q^{\frac{1}{2}(\alpha+\beta+2)})}% {}P_{n-1}^{(\alpha+1,\beta+1)}(x|q)}}}

Proof

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Symbols List

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

Bibliography

Equation in Section 14.10 of KLS.

URL links

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