:24

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∫ - 1 1 S m ( α , β ) ⁡ ( x ) S n ( α , β ) ⁡ ( x ) | x | 2 ⁢ β + 1 ( 1 - x 2 ) α d x = 0    ( m ≠ n ) fragments superscript subscript 1 1 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 𝑚 𝑥 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 𝑛 𝑥 | x superscript | 2 𝛽 1 superscript fragments ( 1 superscript 𝑥 2 ) 𝛼 d x 0 italic-   fragments ( m n ) {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}S^{(\alpha,\beta)}_{m}% \left(x\right)S^{(\alpha,\beta)}_{n}\left(x\right)|x|^{2\beta+1}(1-x^{2})^{% \alpha}dx=0\qquad(m\neq n)}}}

Proof

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Symbols List

∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
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

Bibliography

Equation in Section of KLS.

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