Title: On some integrals over the product of three Legendre functions
Abstract: The definite integrals $\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2}[P_{\nu}(x)]^{3}\, \mathrm{d}x$ , $\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2} [P_{\nu}(x)]^{2}P_{\nu}(-x)\, \mathrm{d}x$ , $\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2}[P_{\nu+1}(x)]^{3}\,\mathrm{d}x$ , and $\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2} [P_{\nu+1}(x)]^{2}P_{\nu +1}(-x)\,\mathrm{d}x $ are evaluated in closed form, where P ν is the Legendre function of degree ν, and $\operatorname{Re}\nu>-1$ . Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.