Difference between revisions of "Orthogonality of Gegenbauer C on (-1,1)"
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(Created page with "==Theorem== The following formula holds: $$\displaystyle\int_{-1}^1 (1-x^2)^{\lambda-\frac{1}{2}} C_n^{\lambda}(x)C_m^{\lambda}(x) \mathrm{d}x = \left\{ \begin{array}{ll} 0, &...") |
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Latest revision as of 23:45, 19 December 2017
Theorem
The following formula holds: $$\displaystyle\int_{-1}^1 (1-x^2)^{\lambda-\frac{1}{2}} C_n^{\lambda}(x)C_m^{\lambda}(x) \mathrm{d}x = \left\{ \begin{array}{ll} 0, & \quad m \neq n \\ \dfrac{2^{1-2\lambda} \pi\Gamma(n+2\lambda)}{(n+\lambda)(\Gamma(\lambda))^2\Gamma(n+1)}, & \quad m=n, \end{array} \right.$$ where $C_n^{\lambda}$ denotes Gegenbauer C, $\pi$ denotes pi, and $\Gamma$ denotes gamma.
