Formula:KLS:14.10:33

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P n ( α , β ) ⁡ ( x | q - 1 ) = q - n ⁢ α ⁢ P n ( α , β ) ⁡ ( x | q )   and   P n ( α , β ) ⁡ ( x ; q - 1 ) = q - n ⁢ ( α + β ) ⁢ P n ( α , β ) ⁡ ( x ; q ) formulae-sequence continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑞 1 superscript 𝑞 𝑛 𝛼 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 and continuous-q-Jacobi-Rahman-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑞 1 superscript 𝑞 𝑛 𝛼 𝛽 continuous-q-Jacobi-Rahman-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q^{% -1}\right)=q^{-n\alpha}P^{(\alpha,\beta)}_{n}\!\left(x|q\right)\quad\textrm{% and}\quad P^{(\alpha,\beta)}_{n}\!\left(x;q^{-1}\right)=q^{-n(\alpha+\beta)}P^% {(\alpha,\beta)}_{n}\!\left(x;q\right)}}}

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
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-Rahman-polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqJacobiRahman

Bibliography

Equation in Section 14.10 of KLS.

URL links

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