On Some Integrals Over the Product of Three Legendre Functions
arXiv:1304.1606 · doi:10.1007/s11139-013-9504-0
Abstract
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 is the Legendre function of degree , and . Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.
Revised according to referee's report, 9 pages