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${\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x\right)=\frac{{\left(\alpha+\beta+1\right)_{m}}}{{\left(\beta+1\right)_{m}}}P^{(\alpha,\beta)}_{m}\left(2x^{2}-1\right)S^{(\alpha,\beta)}_{2m+1}\left(x\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{n}}}}$ : Generalized Gegenbauer polynomial : http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}$ : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3

Bibliography

Equation in Section of KLS.

URL links

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formula in Jacobi: Special cases

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