On Some Integrals Over the Product of Three Legendre Functions

The definite integrals $ \int_{-1}^1(1-x^2)^{(ν-1)/2}[P_ν(x)]^3\D x$, $ \int_{-1}^1(1-x^2)^{(ν-1)/2}[P_ν(x)]^2P_ν(-x)\D x$, $\int_{-1}^1x(1-x^2)^{(ν-1)/2}[P_{ν+1}(x)]^3\D x $ and $\int_{-1}^1x(1-x^2)^{(ν-1)/2}[P_{ν+1}(x)]^2P_{ν+1}(-x)\D x $ are evaluated in closed form, where $P_ν$ is the Legendre function of degree $ν$, and $\Rν>-1$. Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.

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