:14

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C 2 ⁢ n α + 1 2 ⁡ ( x ) C 2 ⁢ n α + 1 2 ⁡ ( 1 ) = P 2 ⁢ n ( α , α ) ⁡ ( x ) P 2 ⁢ n ( α , α ) ⁡ ( 1 ) ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 1 Jacobi-polynomial-P 𝛼 𝛼 2 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛼 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n}% \left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n}\left(1\right)}=\frac{P^{(\alpha,% \alpha)}_{2n}\left(x\right)}{P^{(\alpha,\alpha)}_{2n}\left(1\right)}}}}

Proof

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Symbols List

C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
P n ( α , β ) subscript superscript 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}  : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3

Bibliography

Equation in Section of KLS.

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