:17

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