Formula:KLS:14.10:32

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C 2 n + 1 ( x ; q λ | q ) = ( q λ , - 1 ; q ) n + 1 ( q 1 2 , - q 1 2 ; q ) n + 1 q - 1 2 n x P n ( λ - 1 2 , 1 2 ) ( 2 x 2 - 1 ; q ) continuous-q-ultraspherical-Rogers-polynomial 2 𝑛 1 𝑥 superscript 𝑞 𝜆 𝑞 q-Pochhammer-symbol superscript 𝑞 𝜆 1 𝑞 𝑛 1 q-Pochhammer-symbol superscript 𝑞 1 2 superscript 𝑞 1 2 𝑞 𝑛 1 superscript 𝑞 1 2 𝑛 𝑥 continuous-q-Jacobi-Rahman-polynomial-P 𝜆 1 2 1 2 𝑛 2 superscript 𝑥 2 1 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{2n+1}\!\left(x;q^{\lambda}\,|\,q% \right)=\frac{\left(q^{\lambda},-1;q\right)_{n+1}}{\left(q^{\frac{1}{2}},-q^{% \frac{1}{2}};q\right)_{n+1}}q^{-\frac{1}{2}n}xP^{(\lambda-\frac{1}{2},\frac{1}% {2})}_{n}\!\left(2x^{2}-1;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 : http://dlmf.nist.gov/18.28#E13
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
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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