:22

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S 2 ⁢ m ( α , β ) ⁡ ( x ) := c ⁢ o ⁢ n ⁢ s ⁢ t ⁢ P m ( α , β ) ⁡ ( 2 ⁢ x 2 - 1 ) , S 2 ⁢ m + 1 ( α , β ) ⁡ ( x ) : : assign generalized-Gegenbauer-polynomial-S 𝛼 𝛽 2 𝑚 𝑥 c 𝑜 𝑛 𝑠 𝑡 Jacobi-polynomial-P 𝛼 𝛽 𝑚 2 superscript 𝑥 2 1 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 2 𝑚 1 𝑥 absent {\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x% \right):=\mathrm{c}onstP^{(\alpha,\beta)}_{m}\left(2x^{2}-1\right),\qquad S^{(% \alpha,\beta)}_{2m+1}\left(x\right):}}}

Proof

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Symbols List

S n ( α , β ) subscript superscript 𝑆 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{n}}}}  : Generalized Gegenbauer polynomial : http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer
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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